Part four Appendices Appendix A. Locally ﬁnite homology with local coeﬃcients Appendix B. A brief history of end spaces Appendix C. A brief history of wrapping up References Index

Introduction

We take ‘complex’ to mean both a CW (or simplicial) complex in topology and a chain complex in algebra. An ‘end’ of a complex is a subcomplex with a particular type of inﬁnite behaviour, involving non-compactness in topology and inﬁnite generation in algebra. The ends of manifolds are of greatest interest; we regard the ends of CW and chain complexes as tools in the investigation of manifolds and related spaces, such as stratiﬁed sets. The interplay of the topological properties of the ends of manifolds, the homotopy theoretic properties of the ends of CW complexes and the algebraic properties of the ends of chain complexes has been an important theme in the classiﬁcation theory of high dimensional manifolds for over 35 years. However, the gaps in the literature mean that there are still some loose ends to wrap up! Our aim in this book is to present a systematic exposition of the various types of ends relevant to manifold classiﬁcation, closing the gaps as well as obtaining new results. The book is intended to serve both as an account of the existing applications of ends to the topology of high dimensional manifolds and as a foundation for future developments. We assume familiarity with the basic language of high dimensional manifold theory, and the standard applications of algebraic K- and L-theory to manifolds, but otherwise we have tried to be as self contained as possible. The algebraic topology of ﬁnite CW complexes suﬃces for the combinatorial topology of compact manifolds. However, in order to understand the difference between the topological and combinatorial properties it is necessary to deal with inﬁnite CW complexes and non-compact manifolds. The classic cases include the Hauptvermutung counterexamples of Milnor [96], the topological invariance of the rational Pontrjagin classes proved by Novikov [103], the topological manifold structure theory of Kirby and Siebenmann [84], and the topological invariance of Whitehead torsion proved by Chapman [22]. The algebraic and geometric topology of non-compact manifolds has been a prominent feature in much of the recent work on the Novikov ix

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conjectures – see Ferry, Ranicki and Rosenberg [59] for a survey. (In these applications the non-compact manifolds arise as the universal covers of aspherical compact manifolds, e.g. the Euclidean space Ri covering the torus T i = S 1 × S 1 × . . . × S 1 = B Zi .) In fact, many current developments in topology, operator theory, diﬀerential geometry, hyperbolic geometry, and group theory are concerned with the asymptotic properties of non-compact manifolds and inﬁnite groups – see Gromov [65], Connes [33] and Roe [135] for example. What is an end of a topological space? Roughly speaking, an end of a non-compact space W is a component of W \K for arbitrarily large compact subspaces K ⊆ W . More precisely : Deﬁnition 1. (i) A neighbourhood of an end in a non-compact space W is a subspace U ⊂ W which contains a component of W \K for a non-empty compact subspace K ⊂ W . (ii) An end of W is an equivalence class of sequences of connected open neighbourhoods W ⊃ U1 ⊃ U2 ⊃ . . . such that
∞

The theory of ends was initiated by Freudenthal [61] in connection with topological groups. The early applications of the theory concerned the ends of open 3-dimensional manifolds, and the ends of discrete groups (which are the ends of the universal covers of their classifying spaces). We are especially interested in the ends of manifolds which are ‘tame’, and in extending the notion of tameness to other types of ends. An end of a manifold is tame if it has a system of neighbourhoods satisfying certain strong restrictions on the fundamental group and chain homotopy type. Any non-compact space W can be compactiﬁed by adding a point at inﬁnity, W ∞ = W ∪ {∞}. A manifold end is ‘collared’ if it can be compactiﬁed by a manifold, i.e. if the point at inﬁnity can be replaced by a closed manifold boundary, allowing the end to be identiﬁed with the interior of a compact

Introduction

xi

manifold with boundary. A high dimensional tame manifold end can be collared if and only if an algebraic K-theory obstruction vanishes. The theory of tame ends has found wide application in the surgery classiﬁcation theory of high dimensional compact manifolds and stratiﬁed spaces, and in the related controlled topology and algebraic K- and L-theory. Example 2. Let K be a connected compact space. (i) K × [0, ∞) has one end , with connected open neighbourhoods Ui = K × (i, ∞) ⊂ K × [0, ∞) , such that π1 ( ) = π1 (K). (ii) K × R has two ends
+, −,

Deﬁnition 4. An end of an open n-dimensional manifold W can be collared if it has a neighbourhood of the type M ×[0, ∞) ⊂ W for a connected closed (n − 1)-dimensional manifold M . Example 5. (i) An open n-dimensional manifold with one end is (homeomorphic to) the interior of a closed n-dimensional manifold if and only if can be collared. More generally, if W is an open n-dimensional manifold with compact boundary ∂W and one end , then there exists a compact n-dimensional cobordism (L; ∂W, M ) with L\M homeomorphic to W rel ∂W if and only if can be collared. (ii) If (V, ∂V ) is a compact n-dimensional manifold with boundary then for any x ∈ V \∂V the complement W = V \{x} is an open n-dimensional manifold with a collared end and ∂W = ∂V , with a neighbourhood M ×[0, ∞) ⊂ W for M = S n−1 . The one-point compactiﬁcation of W is W ∞ = V . The compactiﬁcation of W provided by (i) is L = cl(V \Dn ), for any neighbourhood Dn ⊂ V \∂V of x, with (L; ∂W, M ) = (W ∪ S n−1 ; ∂V, S n−1 ). Stallings [154] used engulﬁng to prove that if W is a contractible open n-dimensional P L manifold with one end such that π1 ( ) = {1} and n ≥ 5 then W is P L homeomorphic to Rn – in particular, the end can be collared. Let (W, ∂W ) be an open n-dimensional manifold with compact boundary and one end . Making a proper map d : (W, ∂W )−→([0, ∞), {0}) transverse regular at some t ∈ (0, ∞) gives a decomposition of (W, ∂W ) as (W, ∂W ) = (L; ∂W, M ) ∪M (N, M ) with (L; ∂W, M ) = d−1 ([0, t]; {0}, {t}) a compact n-dimensional cobordism and N = d−1 [t, ∞) non-compact. The end can be collared if and only if N can be chosen such that there exists a homeomorphism N ∼ M × [0, ∞) = rel M = M × {0}, in which case L\M ∼ L ∪M ×{0} M × [0, ∞) ∼ W = = rel ∂W . In terms of Morse theory : it is possible to collar if and only if (W, ∂W ) admits a proper Morse function d with only a ﬁnite number of critical points. Browder, Levine and Livesay [14] used codimension 1 surgery on M ⊂ W to show that if π1 (W ) = π1 ( ) = {1} and n ≥ 6 then can be collared if and only if the homology groups H∗ (W ) are ﬁnitely generated (with Hr (W ) = 0 for all but ﬁnitely many values of r). Siebenmann [140] combined codimension 1 surgery with the ﬁniteness obstruction theory of Wall [163] for ﬁnitely dominated spaces, proving that in dimensions ≥ 6 a tame manifold end can be collared if and only if an algebraic K-theory obstruction vanishes. Deﬁnition 6. A space X is ﬁnitely dominated if there exist a ﬁnite CW complex K and maps f : X−→K, g : K−→X with gf 1 : X−→X.

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xiii

Example 7. Any space homotopy equivalent to a ﬁnite CW complex is ﬁnitely dominated. Example 8. A connected CW complex X with π1 (X) = {1} is ﬁnitely dominated if and only if H∗ (X) is ﬁnitely generated, if and only if X is homotopy equivalent to a ﬁnite CW complex. For non-simply-connected X the situation is more complicated : Theorem 9. (Wall [163, 164]) A connected CW complex X is ﬁnitely dominated if and only if π1 (X) is ﬁnitely presented and the cellular Z[π1 (X)]module chain complex C(X) of the universal cover X is chain equivalent to a ﬁnite f.g. projective Z[π1 (X)]-module chain complex P . The reduced projective class of a ﬁnitely dominated X
∞

[X] = [P ] =

(−)r [Pr ] ∈ K0 (Z[π1 (X)])

r=0

is the ﬁniteness obstruction of X, such that [X] = 0 if and only if X is homotopy equivalent to a ﬁnite CW complex. Deﬁnition 10. An end of an open manifold W is tame if it admits a sequence W ⊃ U1 ⊃ U2 ⊃ . . . of ﬁnitely dominated neighbourhoods with
∞

cl(Ui ) = ∅ , π1 (U1 ) = π1 (U2 ) = . . . = π1 ( ) .
i=1

Example 11. If an end of an open manifold W can be collared then it is tame : if M × [0, ∞) ⊂ W is a neighbourhood of then the open neighbourhoods W ⊃ U1 = M × (1, ∞) ⊃ U2 = M × (2, ∞) ⊃ . . . satisfy the conditions of Deﬁnition 10, with cl(Ui ) = M × [i, ∞), π1 ( ) = π1 (M ). Tameness is a geometric condition which ensures stable (as opposed to wild) behaviour in the topology at inﬁnity of a non-compact space W . The fundamental example is W = K × [0, ∞) for a compact space K, in which the topology at inﬁnity is that of K. Theorem 12. (Siebenmann [140]) A tame end of an open n-dimensional manifold W has a reduced projective class invariant, the end obstruction [ ] = lim [Ui ] ∈ K0 (Z[π1 ( )]) = lim K0 (Z[π1 (Ui )]) ← − ← −
i i

such that [ ] = 0 if (and for n ≥ 6 only if )

can be collared.

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Even if a tame manifold end can be collared, the collarings need not be unique. The various collarings of a tame end in an open manifold W of dimension ≥ 6 with [ ] = 0 ∈ K0 (Z[π1 ( )]) are classiﬁed by the Whitehead group W h(π1 ( )) : if M × [0, ∞), M × [0, ∞) ⊂ W are two collar neighbourhoods of then for suﬃciently large t ≥ 0 there exists an h-cobordism (N ; M, M ) between M × {0} and M × {t} ⊂ W , with M × [0, ∞) = N ∪M
×{t}

M × [t, ∞) ⊂ W .

By the s-cobordism theorem (N ; M, M ) is homeomorphic to the product M × (I; {0}, {1}) if and only if τ (M N ) = 0 ∈ W h(π1 ( )). The nonuniqueness of collarings of P L manifold ends was used by Milnor [96] in the construction of homeomorphisms of compact polyhedra which are not homotopic to a P L homeomorphism, disproving the Hauptvermutung for compact polyhedra. The end obstruction theory played an important role in the disproof of the manifold Hauptvermutung by Casson and Sullivan (Ranicki [131]) – the manifold case also requires surgery and L-theory. Quinn [114, 115, 116] developed a controlled version of the Siebenmann end obstruction theory, and applied it to stratiﬁed spaces. (See Ranicki and Yamasaki [132] for a treatment of the controlled ﬁniteness obstruction, and Connolly and Vajiac [34] for an end theorem for stratiﬁed spaces.) The tameness condition of Deﬁnition 10 for manifold ends was extended by Quinn to stratiﬁed spaces, distinguishing two tameness conditions for ends of non-compact spaces, involving maps pushing forward along the end and in the reverse direction. We shall only consider the two-stratum case of a one-point compactiﬁcation, with the lower stratum the point at inﬁnity. In Chapters 7, 8 we state the deﬁnitions of forward and reverse tameness. The original tameness condition of Siebenmann [140] appears in Chapter 8 as reverse π1 -tameness, so called since it is a combination of reverse tameness and π1 -stability. In general, forward and reverse tameness are independent of each other, but for π1 -stable manifold ends with ﬁnitely presented π1 ( ) the two kinds of tameness are equivalent by a kind of Poincar´ duality. e Deﬁnition 13. (Quinn [116]) The end space e(W ) of a space W is the space of proper paths ω : [0, ∞)−→W . We refer to Appendix B for a brief history of end spaces. The end space e(W ) is a homotopy model for the ‘space at inﬁnity’ of W , playing a role similar to the ideal boundary in hyperbolic geometry. The topology at inﬁnity of a space W is the inverse system of complements of compact subspaces (i.e. cocompact subspaces or neighbourhoods of inﬁnity) of W , which are the open neighbourhoods of the point ∞ in the one-point ∞ compactiﬁcation W ∞ = W ∪ {∞}. The homology at inﬁnity H∗ (W ) is

lf and H∗ (W ) = H∗ (W ∞ , {∞}) for reasonable W . The end space e(W ) is the ‘link of inﬁnity in W ∞ ’. There is a natural passage from the algebraic topology at inﬁnity of W to the algebraic topology of e(W ), which is a one∞ to-one correspondence for forward tame W , with H∗ (e(W )) = H∗ (W ).

are closed neighbourhoods of the two ends. In Chapter 15 we shall prove that the two ends of M are tame, with homotopy equivalences e(M )
+

e(M )

−

M .

The problem of deciding if an open manifold is the interior of a compact manifold with boundary is closely related to the problem of deciding if a compact manifold M ﬁbres over S 1 , i.e. if a map c : M −→S 1 is homotopic to the projection of a ﬁbre bundle. In the ﬁrst instance, it is necessary for (M, c) to be a band :

with [ ] = 0 if and only if can be collared, in which case there exists a compact n-dimensional cobordism (K; ∂W, L) with a rel ∂ homeomorphism (K\L, ∂W ) ∼ (W, ∂W ) = and a homeomorphism (K; ∂W, L) × S 1 ∼ (N ; ∂W × S 1 , M ) = (N as in (i)), and (M, c) ﬁbres over S 1 with M ∼ L × S 1 and M ∼ L × R. = = A CW complex X is ﬁnitely dominated if and only if X × S 1 is homotopy equivalent to a ﬁnite CW complex, by a result of M. Mather [91]. A manifold end of dimension ≥ 6 is tame if and only if × S 1 can be collared – this was already proved by Siebenmann [140], but the wrapping up procedure of Theorem 19 actually gives a canonical collaring of × S 1 . In principle, Theorem 19 could be proved using the canonical regular neighbourhood theory of Siebenmann [148] and Siebenmann, Guillou and H¨hl [149]. We prefer to give a more elementary approach, using a combia nation of the geometric, homotopy theoretic and algebraic methods which have been developed in the last 25 years to deal with non-compact spaces. While the wrapping up construction has been a part of the folklore, the new aspect of our approach is that we rely on the end space and the extensively developed theory of manifold approximate ﬁbrations rather than ad hoc engulﬁng methods. An approximate ﬁbration is a map with an approximate lifting property. (Of course, manifold approximate ﬁbration theory relies on engulﬁng, but we prefer to subsume the details of the engulﬁng in the theory.) We do not assume previous acquaintance with approximate ﬁbrations and engulﬁng. The proof of Theorem 19 occupies most of Parts One and Two (Chapters 1–20). There are three main steps in passing from a tame end of W to the wrapping up band (M, c) such that the inﬁnite cyclic cover M ⊆ W is a neighbourhood of : (i) in Chapter 9 we show that tameness conditions on a space W imply that the end space e(W ) is ﬁnitely dominated and that, near inﬁnity, W looks like the product e(W ) × [0, ∞) ; (ii) in Chapter 16 we use (i) to prove that every tame manifold end of

Introduction

xix

dimension ≥ 5 has a neighbourhood X which is the total space of a manifold approximate ﬁbration d : X−→R ; (iii) in Chapter 17 we show that for every manifold approximate ﬁbration d : X−→R of dimension ≥ 5 there exists a manifold band (M, c) such that X = M , with a proper homotopy d c : X−→R . The construction in (iii) of the wrapping up (M, c) of (X, d) is by the manifold ‘twist glueing’ due to Siebenmann [145]. The twist glueing construction of manifold bands is extended to the CW category in Chapters 19 and 20. In Part Three (Chapters 21–27) we study the algebraic properties of tame ends in the context of chain complexes over a polynomial extension ring and also in bounded algebra. We obtain an abstract version of Theorem 19, giving a chain complex account of wrapping up : manifold wrapping up induces a CW complex wrapping up, which in turn induces a chain complex wrapping up, and similarly for the various types of twist glueing. In Chapter 15 we introduce the notion of a ribbon (X, d), which is a non-compact space X with a proper map d : X−→R with the homotopy theoretic and homological end properties of the inﬁnite cyclic cover (W , c) of a band (W, c). Ribbons are the homotopy analogues of manifold approximate ﬁbration over R. In Chapter 25 we develop the chain complex versions of CW ribbons as well as algebraic versions of tameness. The study of ends of complexes is particularly relevant to stratiﬁed spaces. A topologically stratiﬁed space is a space X together with a ﬁltration ∅ = X −1 ⊆ X 0 ⊆ X 1 ⊆ . . . ⊆ X n−1 ⊆ X n = X by closed subspaces such that the strata X j \X j−1 are open topological manifolds which satisfy certain tameness conditions and a homotopy link condition. These spaces were ﬁrst deﬁned by Quinn [116] in order to study purely topological stratiﬁed phenomena as opposed to the smoothly stratiﬁed spaces of Whitney [170], Thom [161] and J. Mather [90], and the piecewise linear stratiﬁed spaces of Akin [1] and Stone [159]. Quinn’s paper should be consulted for more precise deﬁnitions. Our results only apply directly to the very special case obtained from the one-point compactiﬁcation W ∞ = X of an open manifold W , regarded as a ﬁltered space by X 0 = {∞} ⊆ W ∞ = X. Then X is a topologically stratiﬁed space with two strata if and only if W is tame. (The general case requires controlled versions of our results.) Earlier, Siebenmann [147] had studied a class of topologically stratiﬁed spaces called locally conelike stratiﬁed spaces. The one-point compactiﬁcation of an open manifold W with one end is locally conelike stratiﬁed if and only if the end of W can be collared. Hence, Quinn’s stratiﬁed spaces are much more general than Siebenmann’s. The

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conditions required of topologically stratiﬁed spaces by Quinn are designed to imply that strata have neighbourhoods which are homotopy equivalent to mapping cylinders of ﬁbrations, whereas in the classical cases the strata have neighbourhoods which are homeomorphic to mapping cylinders of bundle projections in the appropriate category : ﬁbre bundle projections in the smooth case, block bundle projections in the piecewise linear case. Strata in Siebenmann’s locally conelike stratiﬁed spaces have neighbourhoods which are locally homeomorphic to mapping cylinders of ﬁbre bundle projections, but not necessarily globally. A stratiﬁed homotopy equivalence is a homotopy equivalence in the stratiﬁed category (maps must preserve strata, not just the ﬁltration). In the special case of one-point compactiﬁcations, stratiﬁed homotopy equivalences (W ∞ , {∞})−→(V ∞ , {∞}) are exactly the proper homotopy equivalences W −→V . Weinberger [166] has developed a stratiﬁed surgery theory which classiﬁes topologically stratiﬁed spaces up to stratiﬁed homotopy equivalence in the same sense that classical surgery theory classiﬁes manifolds up to homotopy equivalence. Weinberger outlines two separate proofs of his theory. The ﬁrst proof [166, pp. 182–188] involves stabilizing a stratiﬁed space by crossing with high dimensional tori in order to get a nicer stratiﬁed space which is amenable to the older stratiﬁed surgery theory of Browder and Quinn [15]. The obstruction to codimension i destabilization involves the codimension i lower K-group K1−i (Z[π]) ⊆ W h(π × Zi ). (Example 18 and Theorem 19 treat the special case i = 1.) The second proof outlined in [166, Remarks p. 189] uses more directly the existence of appropriate tubular neighbourhoods of strata called teardrop neighbourhoods. These neighbourhoods were shown to exist in the case of two strata by Hughes, Taylor, Weinberger and Williams [76] and in general by Hughes [74]. In 16.13 we give a complete proof of the existence of teardrop neighbourhoods in the special case of the topologically stratiﬁed space (W ∞ , {∞}) determined by an open manifold W with a tame end. The result asserts that W contains an open cocompact subspace X ⊆ W which admits a manifold approximate ﬁbration X−→R. In the more rigid smoothly stratiﬁed spaces, the tubular neighbourhoods would be given by a genuine ﬁbre bundle projection. The point is that Quinn’s deﬁnition gives information on the neighbourhoods of strata only up to homotopy. The existence of teardrop neighbourhoods means there is a much stronger geometric structure given in terms of manifold approximate ﬁbrations. We use the theory of manifold approximate ﬁbrations to perform geometric wrapping up constructions. This is analogous to Weinberger’s second approach to stratiﬁed surgery, in which teardrop neighbourhoods of strata are used in order to be able to draw on manifold approximate ﬁbration theory rather than stabilization and destabilization. We expect that the

Introduction

xxi

general theory of teardrop neighbourhoods will likewise allow generalizations of the wrapping up construction to arbitrary topologically stratiﬁed spaces, using the homotopy theoretic and algebraic properties of the ribbons introduced in this book. Such a combination of geometry, homotopy theory and algebra will be necessary to fully understand the algebraic Kand L-theory of stratiﬁed spaces. This book grew out of research begun in 1990–91 when the ﬁrst-named author was a Fulbright Scholar at the University of Edinburgh. We have received support from the National Science Foundation (U.S.A.), the Science and Engineering Research Council (U.K.), the European Union K-theory Initiative under Science Plan SCI–CT91–0756, the Vanderbilt University Research Council, and the Mathematics Departments of Vanderbilt University and the University of Edinburgh. We have beneﬁted from conversations with Stratos Prassidis and Bruce Williams. A The book was typeset in TEX, with the diagrams created using the L MSTEX, PICTEX and X -pic packages. Y Errata (if any) to this book will be posted on the WWW Home Page http://www.maths.ed.ac.uk/people/aar

Chapter summaries

Part One, Topology at inﬁnity, is devoted to the basic theory of the general, geometric and algebraic topology at inﬁnity of non-compact spaces. Various models for the topology at inﬁnity are introduced and compared. Chapter 1, End spaces, begins with the deﬁnition of the end space e(W ) of a non-compact space W . The set of path components π0 (e(W )) is shown to be in one-to-one correspondence with the set of ends of W (in the sense of Deﬁnition 1 above) for a wide class of spaces. Chapter 2, Limits, reviews the basic constructions of homotopy limits and colimits of spaces, and the related inverse, direct and derived limits of groups and chain complexes. The end space e(W ) is shown to be weak homotopy equivalent to the homotopy inverse limit of cocompact subspaces of W and the homotopy inverse limit is compared to the ordinary inverse limit. ∞ The ‘fundamental group at inﬁnity’ π1 (W ) of W is deﬁned and compared to π1 (e(W )). Chapter 3, Homology at inﬁnity, contains an account of locally ﬁnite singular homology, which is the homology based on inﬁnite chains. The ho∞ mology at inﬁnity H∗ (W ) of a space W is the diﬀerence between ordinary lf singular homology H∗ (W ) and locally ﬁnite singular homology H∗ (W ). Chapter 4, Cellular homology, reviews locally ﬁnite cellular homology, although the technical proof of the equivalence with locally ﬁnite singular homology is left to Appendix A. Chapter 5, Homology of covers, concerns ordinary and locally ﬁnite singular and cellular homology of the universal cover (and other covers) W of W . The version of the Whitehead theorem for detecting proper homotopy equivalences of CW complexes is stated. Chapter 6, Projective class and torsion, recalls the Wall ﬁniteness obstruction and Whitehead torsion. A locally ﬁnite ﬁniteness obstruction is introduced, which is related to locally ﬁnite homology in the same way that the Wall ﬁniteness obstruction is related to ordinary homology, and the diﬀerence between the two obstructions is related to homology at inﬁnity. xxii

Chapter summaries

xxiii

Chapter 7, Forward tameness, concerns a tameness property of ends, which is stated in terms of the ability to push neighbourhoods towards inﬁnity. It is proved that for forward tame W the singular chain complex of the end space e(W ) is chain equivalent to the singular chain complex at inﬁnity of W , and that the homotopy groups of e(W ) are isomorphic to the inverse limit of the homotopy groups of cocompact subspaces of W . There is a related concept of forward collaring. Chapter 8, Reverse tameness, deals with the other tameness property of ends, which is stated in terms of the ability to pull neighbourhoods in from inﬁnity. It is closely related to ﬁnite domination properties of cocompact subspaces of W . There is a related concept of reverse collaring. Chapter 9, Homotopy at inﬁnity, gives an account of proper homotopy theory at inﬁnity. It is shown that the homotopy type of the end space, the two types of tameness, and other end phenomena are invariant under proper homotopy equivalences at inﬁnity. It is also established that in most cases of interest a space W is forward and reverse tame if and only if W is bounded homotopy equivalent at ∞ to e(W ) × [0, ∞), in which case e(W ) is ﬁnitely dominated. Chapter 10, Projective class at inﬁnity, introduces two ﬁniteness obstructions which the two types of tameness allow to be deﬁned. The ﬁniteness obstruction at inﬁnity of a reverse tame space is an obstruction to reverse collaring. Likewise, the locally ﬁnite ﬁniteness obstruction at inﬁnity of a forward tame space is an obstruction to forward collaring. For a space W which is both forward and reverse tame, the end space e(W ) is ﬁnitely dominated and its Wall ﬁniteness obstruction is the diﬀerence of the two ﬁniteness obstructions at inﬁnity. It is also proved that for a manifold end forward and reverse tameness are equivalent under certain fundamental group conditions. Chapter 11, Inﬁnite torsion, contains an account of the inﬁnite simple homotopy theory of Siebenmann for locally ﬁnite CW complexes. The inﬁnite Whitehead group of a forward tame CW complex is described algebraically as a relative Whitehead group. The inﬁnite torsion of a proper homotopy equivalence is related to the locally ﬁnite ﬁniteness obstruction at inﬁnity. A CW complex W is forward (resp. reverse) tame if and only if W × S 1 is inﬁnite simple homotopy equivalent to a forward (resp. reverse) collared CW complex. Chapter 12, Forward tameness is a homotopy pushout, deals with Quinn’s characterization of forward tameness for a σ-compact metric space W in terms of a homotopy property, namely that the one-point compactiﬁcation W ∞ is the homotopy pushout of the projection e(W )−→W and e(W )−→ {∞}, or equivalently that W ∞ is the homotopy coﬁbre of e(W )−→W . Part Two, Topology over the real line, concerns spaces W with a proper map d : W −→R.

xxiv

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Chapter 13, Inﬁnite cyclic covers, proves that a connected inﬁnite cyclic + − cover W of a connected compact AN R W has two ends W , W , and + establishes a duality between the two types of tameness : W is forward − tame if and only if W is reverse tame. A similar duality holds for forward and reverse collared ends. Chapter 14, The mapping torus, works out the end theory of inﬁnite cyclic covers of mapping tori. Chapter 15, Geometric ribbons and bands, presents bands and ribbons. It is proved that (M, c : M −→S 1 ) with M a ﬁnite CW complex deﬁnes a band (i.e. the inﬁnite cyclic cover M = c∗ R of M is ﬁnitely dominated) if + − and only if the ends M , M are both forward tame, or both reverse tame. The Siebenmann twist glueing construction of a band is formulated for a ribbon (X, d : X−→R) and an end-preserving homeomorphism h : X−→X. Chapter 16, Approximate ﬁbrations, presents the main geometric tool used in the proof of the uniformization Theorem 19 (every tame manifold end of dimension ≥ 5 has a neighbourhood which is the inﬁnite cyclic cover of a manifold band). It is proved that an open manifold W of dimension ≥ 5 is forward and reverse tame if and only if there exists an open cocompact subspace X ⊆ W which admits a manifold approximate ﬁbration X−→R. Chapter 17, Geometric wrapping up, uses the twist glueing construction with h = 1 : X−→X to prove that the total space X of a manifold approximate ﬁbration d : X−→R is the inﬁnite cyclic cover X = M of a manifold band (M, c). Chapter 18, Geometric relaxation, uses the twist glueing construction with h = covering translation : M −→M to pass from a manifold band (M, c) to an h-cobordant manifold band (M , c ) such that c : M −→S 1 is a manifold approximate ﬁbration. Chapter 19, Homotopy theoretic twist glueing, and Chapter 20, Homotopy theoretic wrapping up and relaxation, extend the geometric constructions for manifolds in Chapters 17 and 18 to CW complex bands and ribbons. Constructions in this generality serve as a bridge to the algebraic theory of Part Three. Moreover, it is shown that any CW ribbon is inﬁnite simple homotopy equivalent to the inﬁnite cyclic cover of a CW band, thereby justifying the concept. Part Three, The algebraic theory, translates most of the geometric, homotopy theoretic and homological constructions of Parts One and Two into an appropriate algebraic context, thereby obtaining several useful algebraic characterizations. Chapter 21, Polynomial extensions, gives background information on chain complexes over polynomial extension rings, motivated by the fact that the cellular chain complex of an inﬁnite cyclic cover of a CW complex is deﬁned over a Laurent polynomial extension.

Chapter summaries

xxv

Chapter 22, Algebraic bands, discusses chain complexes over Laurent polynomial extensions which have the algebraic properties of cellular chain complexes of CW complex bands. Chapter 23, Algebraic tameness, develops the algebraic analogues of forward and reverse tameness for chain complexes over polynomial extensions. This yields an algebraic characterization of forward (and reverse) tameness for an end of an inﬁnite cyclic cover of a ﬁnite CW complex. End complexes are also deﬁned in this algebraic setting. Chapter 24, Relaxation techniques, contains the algebraic analogues of the constructions of Chapters 18 and 20. When combined with the geometry of Chapter 18 this gives an algebraic characterization of manifold bands which admit approximate ﬁbrations to S 1 . Chapter 25, Algebraic ribbons, explores the algebraic analogue of CW ribbons in the context of bounded algebra. The algebra is used to prove that CW ribbons are inﬁnite simple homotopy equivalent to inﬁnite cyclic covers of CW bands. Chapter 26, Algebraic twist glueing, proves that algebraic ribbons are simple chain equivalent to algebraic bands. Chapter 27, Wrapping up in algebraic K- and L-theory, describes the effects of the geometric constructions of Part Two on the level of the algebraic K- and L-groups. Part Four consists of the three appendices : Appendix A, Locally ﬁnite homology with local coeﬃcients, contains a technical treatment of ordinary and locally ﬁnite singular and cellular homology theories with local coeﬃcients. This establishes the equivalence of locally ﬁnite singular and cellular homology for regular covers of CW complexes. Appendix B, A brief history of end spaces, traces the development of end spaces as homotopy theoretic models for the topology at inﬁnity. Appendix C, A brief history of wrapping up, outlines the history of the wrapping up compactiﬁcation procedure.

Part One: Topology at inﬁnity

1
End spaces

Throughout the book it is assumed that AN R spaces are locally compact, separable and metric, and that CW complexes are locally ﬁnite. We start with the end space e(W ) of a space W , which is a homotopy theoretic model for the behaviour at ∞ of W . The homotopy type of e(W ) is determined by the proper homotopy type of W . The set of path components π0 (e(W )) is related to the number of ends of W , and the fundamental group π1 (e(W )) is related to the fundamental group at ∞ of W . Deﬁnition 1.1 The one-point compactiﬁcation of a topological space W is the compact topological space W ∞ = W ∪ {∞} , with open sets : (i) U ⊂ W ∞ for an open subset U ⊆ W , (ii) V ∪ {∞} ⊆ W ∞ for a subset V ⊆ W such that W \V is compact. The topology at inﬁnity of W is the topology of W ∞ at ∞. Deﬁnition 1.2 The end space e(W ) of a space W is the space of paths ω : ([0, ∞], {∞}) −→ (W ∞ , {∞}) such that ω −1 (∞) = {∞}, with the compact-open topology. The end space e(W ) is the homotopy link holink(W ∞ , {∞}) of {∞} in W ∞ in the sense of Quinn [116]. See 1.8 for the connection with the link in the sense of P L topology, and 12.11 for the general deﬁnition of the homotopy link. 1

2

Ends of complexes We refer to Appendix B for a brief history of end spaces.

An element ω ∈ e(W ) can also be viewed as a path ω : [0, ∞)−→W such that ω(t) ‘diverges to ∞’ as t−→∞, meaning that for every compact subspace K ⊂ W there exists N > 0 with ω([N, ∞)) ⊂ W \K. Deﬁnition 1.3 (i) A map of spaces f : V −→W is proper if for each compact subspace K ⊆ W the inverse image f −1 (K) ⊆ V is compact. This is equivalent to the condition that f extends to a map f ∞ : V ∞ −→W ∞ of the one-point compactiﬁcations with f ∞ (∞) = ∞. (ii) A map f : V −→W is a proper homotopy equivalence if it is a proper map which is a homotopy equivalence in the proper category. We refer to Porter [111] for a survey of the applications of proper homotopy theory to ends. The end space e(W ) is called the ‘Waldhausen boundary’ of W in [111, p. 135]. An element ω ∈ e(W ) is a proper map ω : [0, ∞)−→W , which is the same as a path in ω ∞ : [0, ∞]−→W ∞ such that ω ∞ [0, ∞) ⊆ W and ω ∞ (∞) = ∞. Example 1.4 (i) The end space of a compact space W is empty, e(W ) = ∅ , since W ∞ = W ∪{∞} is disconnected and there are no paths ω ∞ : [0, ∞]−→ W ∞ from ω ∞ (0) ∈ W to ω ∞ (∞) = ∞ ∈ W ∞ . The converse is false : the end space of Z is empty, yet Z is not compact. (ii) Let T be a tree, and let v ∈ T be a base vertex. A simple edge path in T is a sequence of adjoining edges e1 , e2 , e3 , . . . (possibly inﬁnite) without repetition. By the simplicial approximation theorem every proper map ω : [0, ∞)−→T is proper homotopic to an inﬁnite simple edge path starting at v. If T has at most a ﬁnite number of vertices of valency > 2 the end space e(T ) is homotopy equivalent to the discrete space with one point for each simple edge path of inﬁnite length starting at v ∈ T . (iii) The end space of R+ = [0, ∞) is contractible, e(R+ ) {pt.} , corresponding to the unique inﬁnite simple edge path starting at 0 ∈ R+ . (iv) The end space of R is such that e(R) S 0 = {+1, −1} , corresponding to the two inﬁnite simple edge paths starting at 0 ∈ R. In dealing with end spaces e(W ), we shall always assume that W is a locally compact Hausdorﬀ space.

with i : W −→W ∞ the inclusion. The space W is ‘forward tame’ if and only if this square is a homotopy pushout rel {∞} – see Chapters 7, 12 for a more detailed discussion. Deﬁnition 1.6 Let (K, L ⊆ K) be a pair of spaces. The space L is collared in K if the inclusion L = L × {0}−→K extends to an open embedding f : L × [0, ∞)−→K. Proposition 1.7 If (K, L) is a compact pair of spaces such that L is collared in K then the end space of the non-compact space W = K\L is such that there is deﬁned a homotopy equivalence L −→ e(W ) ; x −→ (t −→ f x, 1 ) 1+t

with f : L × [0, ∞)−→K an open embedding extending the inclusion L = L × {0}−→K. In other words, if W is a non-compact space with a compactiﬁcation K such that the boundary ∂K = K\W ⊂ K is a compact subset which is collared in K then there is deﬁned a homotopy equivalence e(W ) ∂K .

The homotopy theoretic ‘space at inﬁnity’ e(W ) thus has the homotopy type of an actual space at inﬁnity, provided ∂W is collared in the compactiﬁcation K.

The homotopy link of {∞} in W ∞ is homotopy equivalent to the actual link of x in X. (ii) Let (M, ∂M ) be a compact n-dimensional topological manifold with boundary. The boundary ∂M is collared in M . (In the topological category this was ﬁrst proved by Brown [16]. See Conelly [31] for a more recent proof.) The interior of M is an open n-dimensional manifold W = int(M ) = M \∂M with an open embedding f : ∂M × [0, ∞)−→M extending the inclusion ∂M = ∂M × {0}−→M . The end space of W is such that the map g : ∂M −→ e(W ) ; x −→ (t −→ f x, deﬁnes a homotopy equivalence, with the adjoint of g g : ∂M × [0, ∞) −→ W ; (x, t) −→ g(x)(t) homotopic to f . (iii) In view of (ii) a necessary condition for an open n-dimensional manifold W to be homeomorphic to the interior of a compact n-dimensional manifold with boundary is that the end space e(W ) have the homotopy type of a closed (n − 1)-dimensional manifold. In Chapters 7, 8 we shall be studying geometric tameness conditions on W which ensure that e(W ) is at least a ﬁnitely dominated (n − 1)-dimensional geometric Poincar´ complex. e 1 ) t+1

A subspace V ⊆ W is cocompact if the closure of W \V ⊆ W is compact. For a CW complex W a subcomplex V ⊆ W is coﬁnite if it contains all but ﬁnitely many cells of W . A coﬁnite subcomplex is a cocompact subspace.

6

Ends of complexes
∞

Deﬁnition 1.11 A space W is σ-compact if W =
j=1

Kj

with each Kj compact and Kj ⊆ Kj+1 . In particular, all the AN R’s considered by us are σ-compact, since we are assuming that they are locally compact, separable and metric. It follows from 1.10 that the homotopy type of e(W ) is determined by the proper homotopy type of W . A more general result will be established in 9.4 for a metric space W , that the homotopy type of e(W ) is determined by the ‘proper homotopy type at ∞’ of W . The inclusion of a closed cocompact subspace is a special case of a ‘proper homotopy equivalence at ∞’, and the following result will be used in the proof of 9.4 : Proposition 1.12 If W is a σ-compact metric space and u : V −→W is the inclusion of a closed cocompact subspace then the inclusion of end spaces e(u) : e(V )−→e(W ) is a homotopy equivalence. Proof Since W is a σ-compact metric space, W ∞ and e(W ) are metrizable, and so e(W ) is paracompact. For each ω ∈ e(W ) choose a number tω ∈ [0, ∞) such that ω([tω , ∞)) ⊆ int(V ) . Let U (ω) be an open neighbourhood of ω in e(W ) such that α([tω , ∞)) ⊆ int(V ) (α ∈ U (ω)) . Let {Ui } be a locally ﬁnite reﬁnement of the covering {U (ω) | ω ∈ e(W )} of e(W ), and let {φi } be a partition of unity subordinate to {Ui }. For each i choose ωi ∈ e(W ) such that Ui ⊆ U (ωi ), and let ti = tωi . For each ω ∈ e(W ) let mω = min{ti | φi (ω) = 0} . Note that ω([mω , ∞)) ⊆ intV and
i

In dealing with the number of ends of a space W we shall assume the following standing hypothesis for the rest of this chapter : W is a locally compact, connected, locally connected Hausdorﬀ space (e.g. a locally ﬁnite connected CW complex). In the literature the end space e(W ) has not played as central a role as the ‘ends of W ’ or the ‘number of ends of W ’. Roughly, an end of W should correspond to a path component of e(W ). We now recall these classical notions and their relationship to π0 (e(W )). Deﬁnition 1.14 (Milnor [100]) An end of a space W is a function : {K | K ⊆ W is compact} −→ {X | X ⊆ W } ; K −→ (K) such that : (i) (K) is a component of W \K for each K, (ii) if K ⊆ L, then (L) ⊆ (K). A neighbourhood of is a connected open subset U ⊆ W such that U = (K) for some non-empty compact K ⊆ W . Remark 1.15 (i) For a σ-compact space W the deﬁnition of an end in 1.14 agrees with Deﬁnition 1 in the Introduction. A sequence W ⊇ U1 ⊇ U2 ⊇ . . . of neighbourhoods of an end (in the sense of Deﬁnition 1 of the Introduction) such that
∞ j=1

cl(Uj ) = ∅ determines an end

of W (in the sense of 1.14) as

8

Ends of complexes

follows : for a compact subspace K ⊆ W choose j such that Uj ∩ K = ∅ and let (K) be the component of W \K which contains Uj . On the other hand, if is an end of W and W =
∞ j=1

Kj with each Kj compact and Kj ⊆ Kj+1 ,

then (Kj ) = Uj deﬁnes a sequence of neighbourhoods of an end as above. (ii) A subspace is unbounded if its closure is not compact. Note that if is an end of W , then (K) is unbounded for each compact subspace K ⊆ W . (Otherwise, L = K ∪cl( (K)) would be a compact subspace of W containing K, so (L) ⊆ (K) ⊆ L, contradicting (L) ⊆ W \L.) Deﬁnition 1.16 The number of ends of a locally ﬁnite CW complex W is the least upper bound of the number (which may be inﬁnite) of inﬁnite components of W \V for ﬁnite subcomplexes V ⊂ W . Example 1.17 (i) The real line R has exactly two ends. (ii) The dyadic tree X is the tree embedded in R2 with each vertex of valency 3, with closure the union of X together with a disjoint Cantor set. The dyadic tree has an uncountable number of ends. See Diestel [37] for more information on ends of graphs. An alternative approach to the deﬁnition of an end is to focus attention on the number of ends of a space. Deﬁnition 1.18 (Specker [151], Raymond [134]) The space W has at least k ends if there exists an open subspace V ⊆ W with compact closure cl(V ) such that W \cl(V ) has at least k unbounded components. The space W has (exactly) k ends if W has at least k ends but not at least k + 1 ends. The point set conditions on W imply that if V ⊆ W is an open subspace with compact closure, then W \cl(V ) has at most a ﬁnite number of unbounded components (see Hocking and Young [66, Theorem 3–9, p. 111]). If W has exactly k ends then there exists an open subspace V ⊆ W with compact closure so that W \cl(V ) has exactly k unbounded components. Proposition 1.19 Let k ≥ 0 be an integer. (i) If W has at least k ends in the sense of Deﬁnition 1.14, then W has at least k ends in the sense of Deﬁnition 1.18. (ii) If W is σ-compact and has at least k ends in the sense of Deﬁnition 1.18, then W has at least k ends in the sense of Deﬁnition 1.14. (iii) For W σ-compact, W has exactly k ends in the sense of Deﬁnition 1.14 if and only if W has exactly k ends in the sense of Deﬁnition 1.18. Proof (i) Let 1 , . . . , k be distinct ends of W in the sense of 1.14. For 1 ≤ i < j ≤ k, choose a compact subspace Hij ⊆ W such that i (Hij ) = j (Hij ).

1. End spaces It follows that
i (Hij )

9

∩

j (Hij )

= ∅. Let Hij .
1≤i<j≤k

H =

Since H is compact, there is an open subspace V ⊆ W with compact closure such that H ⊆ V . Then 1 (cl(V )), . . . , k (cl(V )) are unbounded components of W \cl(V ). Since i (cl(V )) ⊆ i (Hij ), these are in fact k distinct components. Thus, W has at least k ends in the sense of 1.18. (ii) We may assume that k ≥ 1, so that W is non-compact. Note that if K ⊆ W is a compact subspace, then W \K has at least one unbounded component. For if V ⊆ W is an open subspace with compact closure such that K ⊆ V , then all but ﬁnitely many components of W \K are contained in V (see Hocking and Young [66, Theorem 3–9, p. 111]). It follows that one of those ﬁnitely many components of W \K must be unbounded. Next, we shall show that if K ⊆ W is a compact subspace and C is an unbounded component of W \K, then there exists an end of W in the sense of 1.14 such that (K) = C. For W can be written as
∞

W =
j=0

Kj

with K0 = K, each Kj compact and Kj ⊆ Kj+1 . Deﬁne as follows. First, let (K) = (K0 ) = C. Then, assuming j ≥ 1 and that (Kj−1 ) has been deﬁned, deﬁne (Kj ) to be one of the unbounded components of cl( (Kj−1 ))\Kj (which exists by the argument above). Finally, for an arbitrary compact subspace H ⊆ W , choose j such that H ⊆ Kj , and deﬁne (H) to be the component of W \H which contains (Kj ). It is easy to verify that is an end of W in the sense of 1.14. Since W has at least k ends in the sense of 1.18, there exists an open subspace V ⊂ W with compact closure such that W \cl(V ) has at least k unbounded components, say C1 , . . . , Ck . Then there exist ends 1 , . . . , k of W in the sense of 1.14 such that j (cl(V )) = Cj for j = 1, . . . , k. (iii) Immediate from (i) and (ii). If a space W is not assumed to be σ-compact, then we shall assume that an end of W refers to an end in the sense of 1.14 unless otherwise stated. Of course, such an end gives rise to an end in the sense of 1.18. Proposition 1.20 A connected space W with exactly k ends can be expressed as
k

for K ⊆ W and i = 1, 2. This shows that W has at least k + 1 ends, a contradiction. Deﬁnition 1.21 The set of ends EW of a space W is the set of ends of W in the sense of Deﬁnition 1.14. Proposition 1.22 (i) The set of path components of the end space e(W ) is related to the set of ends of a space W by the map ηW : π0 (e(W )) −→ EW ; [ω] −→
ω

and the topology on W ∗ is such that neighbourhoods of j are of the form j (K) ∪ { j } for compact subspaces K ⊆ W . (If W has one end this is the one-point compactiﬁcation, W ∗ = W ∞ .) If W has inﬁnitely many ends, the topology on W ∗ is more complicated because the ends are no longer isolated. The compactiﬁcation W ∗ is characterized in [134] by the properties : (i) (ii) (iii) (iv) W ∗ is connected, W is open in W ∗ , W ∗ \W is totally disconnected, if x ∈ W ∗ \W and U is a connected open neighbourhood of x, then U \(W ∗ \W ) is connected.

2
Limits

In this chapter we state the basic constructions and properties of homotopy limits and colimits of spaces, and the related direct and inverse systems of groups. In 2.14 we shall show that the end space e(W ) of a σ-compact space W has the weak homotopy type of the homotopy limit holim Wj of an ←− −− inverse system {Wj | j = 0, 1, 2, . . .} of closed cocompact subspaces Wj ⊆ W with ∅ =
∞ j=0 j

with lim1 denoting the derived limit (2.11). The ‘Mittag–Leﬄer’ and ‘stability’ conditions for an inverse sequence of groups are recalled (2.20) and related to derived limits. The related geometric condition ‘semistability at ∞’ for a space W is interpreted in terms of the end space e(W ) (2.25). We refer to Bousﬁeld and Kan [9] for the general theory of homotopy limits and colimits. Deﬁnition 2.1 The direct limit of a direct system of sets f0 f1 X0 −−→ X1 −−→ X2 −−→ . . . is the quotient of the disjoint union
∞ j

Xj

lim Xj = − →
j

Xj
j=0

(xj+1 = fj (xj )) .

13

14

Ends of complexes

Proposition 2.2 (i) For any set X and any sequence of subsets X0 ⊆ X1 ⊆ X2 ⊆ . . . ⊆ X the direct limit of the direct system {fj : Xj −→Xj+1 } deﬁned by the inclusions is the union
∞

lim Xj = − →
j

Xj ⊆ X .
j=0

(ii) The direct limit of a direct system of groups {fj : Gj −→Gj+1 } is a group lim Gj . For abelian Gj the direct limit is an abelian group, such that − → j up to isomorphism
∞ ∞

The map K−→e(W ) sends each x ∈ K to the ray joining it to the cone point ∞ ∈ W ∞ = cK. Deﬁnition 2.7 The inverse limit of an inverse system of sets f1 f2 X0 ←−− X1 ←−− X2 ←−− . . . is the subset of the product
j

Xj
∞

lim Xj = {(x0 , x1 , x2 , . . .) ∈ ← −
j

Xj | fj (xj ) = xj−1 } .
j=0

Proposition 2.8 (i) For any set X and any sequence of subsets . . . ⊆ X2 ⊆ X1 ⊆ X0 ⊆ X the inverse limit of the inverse system {fj : Xj −→Xj−1 } deﬁned by the inclusions is the intersection
∞

lim Xj = ← −
j

Xj ⊆ X .
j=0

(ii) The inverse limit of an inverse system of groups {Gj −→Gj−1 } is a group lim Gj . ← −
j

Deﬁnition 2.9 The homotopy inverse limit of an inverse system of spaces I {fj : Xj −→Xj−1 } is the subspace of the product j Xj of the path spaces I Xj
∞

Deﬁnition 2.20 Let {fj : Gj −→Gj−1 | j ≥ 1} be an inverse system of groups. (i) The inverse system is Mittag–Leﬄer if there exists k ≥ 1 such that the morphisms fj | : im(fj+1 ) −→ im(fj ) are onto for all j ≥ k. (ii) The inverse system is stable if there exists k ≥ 1 such that the morphisms fj | : im(fj+1 ) −→ im(fj ) are isomorphisms for all j ≥ k.

As in 1.22 let ηW : π0 (e(W ))−→EW be the function which associates an end of a space W to each path component of the end space e(W ). Deﬁnition 2.23 (i) A σ-compact space W is path-connected at ∞ if every cocompact subspace of W contains a path-connected cocompact subspace of W , or equivalently if there exists a sequence W ⊃ W0 ⊃ W1 ⊃ W2 ⊃ . . . of path-connected cocompact subspaces with cl(Wj ) = ∅.
j

Example 2.24 If (M, ∂M ) is a compact manifold with boundary then each component L of ∂M is collared in the complement W = M \L, so that W ∞ has stable π1 at inﬁnity with π1 (W ) = π1 (L). Proposition 2.25 (i) If W is σ-compact and locally path-connected, then ηW : π0 (e(W ))−→EW is surjective. (ii) W is semistable at ∞ if and only if ηW is injective. (iii) If W is σ-compact and locally path-connected, then W is semistable at ∞ if and only if ηW is bijective. (iv) If W is σ-compact, locally path-connected and semistable at ∞, then W is path-connected at ∞ if and only if π0 (e(W )) = 0. Proof (i) Write
∞

W =
j=0

Kj

with each Kj compact and Kj ⊆ Kj+1 . Let be an end of W (in the sense of 1.14). For each j, choose xj ∈ (Kj ). Then xj , xj+1 ∈ (Kj ) and (Kj ) is a component of an open subset of a locally path-connected space. Hence, (Kj ) is path-connected, so there is a map ωj : [j, j + 1] −→ (Kj ) with ωj (j) = xj and ωj (j + 1) = xj+1 . Then the ωj ’s amalgamate to deﬁne a proper map ω : [0, ∞)−→W with ηW ([ω]) = . (ii) Immediate from the Deﬁnition 2.23. (iii) follows from (i) and (ii). (iv) Since ηW is bijective, we need to show that W is path-connected at ∞ if and only if W has exactly one end. Suppose that W is path-connected at ∞ and that 1 , 2 are distinct ends of W . Then there is a compact subspace K ⊆ W such that 1 (K) = 2 (K), and so 1 (K) ∩ 2 (K) = ∅. Let X be a path-connected cocompact subspace of W \K. Since 1 (K), 2 (K) are unbounded
1 (K)

∩X = ∅ =

2 (K)

∩ X.

But X must be contained in exactly one of the components of W \K, a contradiction. On the other hand, if is the only end of W , then for every cocompact subspace X ⊆ W , (cl(W \X)) must be a path-connected cocompact subspace of W . Example 2.26 Jacob’s ladder can be realized as the subspace X ⊂ R2 deﬁned by X = {(x, y) | x = 0, 1 , y ≥ 0} ∪ {(x, n) | 0 ≤ x ≤ 1 , n = 1, 2, 3, . . .} .

26

Ends of complexes

Then X has exactly one end, but π0 (e(X)) is inﬁnite and X is not semistable at ∞.
. . . . . .

Remark 2.27 (i) In 9.5 below it will be proved that if X and Y are proper homotopy equivalent then X is semistable (resp. has stable π1 ) at ∞ if and only if Y is semistable (resp. has stable π1 ) at ∞. (ii) Let W = Kj be a locally path-connected σ-compact space, which is
j

expressed as a union of compact subspaces Kj ⊆ Kj+1 . The complements Wj = W \Kj are cocompact subsets of W such that
∞

(iii) A well-known conjecture states that if W is a ﬁnite connected CW complex, then the universal cover W is semistable at ∞. This is known to be a property of π1 (W ) and has been veriﬁed in many special cases. See Mihalik [93], Mihalik and Tschantz [94]. The homological properties of non-compact spaces are closely related to the localization and completion of rings. Here is a brief account of these constructions, in the special cases of the localization inverting a single element and the completion with respect to a principal ideal. Deﬁnition 2.28 Let A be a ring (associative, with 1) and let s ∈ A be a central non-zero divisor. (i) The localization of A inverting s is the ring A[1/s] with elements the equivalence classes a/sj of pairs (a, sj ) (a ∈ A, j ≥ 0), subject to the equivalence relation (a, sj ) ∼ (b, sk ) if ask = bsj ∈ A , and the usual addition and multiplication of fractions. The localization is (up to isomorphism) the direct limit s s s A[1/s] = lim(A −→ A −→ A −→ A −→ . . .) , − → with an actual isomorphism deﬁned by
∞

∞ The homology at inﬁnity H∗ (W ) of a space W is the proper homotopy invariant given by the diﬀerence between homology H∗ (W ) and locally ﬁlf nite homology H∗ (W ). The extent to which a space W is non-compact is measured in the ﬁrst instance by the failure of the natural maps i : lf H∗ (W )−→H∗ (W ) to be isomorphisms, or equivalently by the extent to ∞ which H∗ (W ) is non-zero. The homology groups of the end space e(W ) ∞ are related to the homology at inﬁnity by morphisms H∗ (e(W ))−→H∗ (W ), which are isomorphisms if W is forward tame (in the sense of Chapter 7).

Locally ﬁnite homology is as important in studying non-compact spaces as ordinary homology is important in dealing with compact spaces. Since there is no elementary account of locally ﬁnite homology in the literature, we provide one here. We shall also investigate the connection between the locally ﬁnite holf mology H∗ (W ) of a space W and the reduced homology of the one-point compactiﬁcation W ∞ H∗ (W ∞ ) = H∗ (W ∞ , {∞}) . In general, these homology groups are not isomorphic – see 3.18 below for an actual example. In 3.16 we identify the singular locally ﬁnite chain complex S lf (W ) of a σ-compact space W with an inverse limit of singular chain complexes, the singular chain complex at inﬁnity S ∞ (W ) with a derived limit of singular chain complexes, and the singular locally ﬁnite homology with the homology of an inverse limit of ordinary singular chain complexes lf involving W ∞ . In Chapter 7 we use 3.16 to prove that H∗ (W ) is isomorphic to H∗ (W ∞ , {∞}) for a forward tame W . 3.16 is used in Appendix A, which relates locally ﬁnite singular and cellular homology to each other. 29

Remark 3.20 Epstein [42] identiﬁes the number of ends (1.14) of a locally ﬁnite CW complex W with the dimension of the real vector space 0 H∞ (W ; R).

4
Cellular homology

It is well-known that the singular homology groups of a CW complex are isomorphic to the cellular homology groups; it is less well documented (and much harder to prove) that the singular locally ﬁnite homology groups of a ‘strongly locally ﬁnite’ CW complex are isomorphic to the cellular locally ﬁnite homology groups. This is stated in 4.7, and is proved in Appendix A. Deﬁnition 4.1 The cellular chain complex of a CW complex W is the free Z-module chain complex C(W ) with chain objects C(W )r = Hr (W (r) , W (r−1) ) =
Ir

Proof There exists a subcomplex Dlf,π (W ) ⊂ S lf,π (W ) such that the natural Z[π]-module chain maps S lf,π (W ) ← Dlf,π (W ) −→ C lf,π (W ) are homology equivalences. See Proposition A.7 in Appendix A. The Whitehead theorem states that a map of connected CW complexes is a homotopy equivalence if and only if it induces isomorphisms of fundamental groups and the homology groups of the universal covers. Farrell, Taylor and Wagoner [51] established a Whitehead theorem in the proper category; roughly speaking, a homotopy equivalence of locally ﬁnite inﬁnite CW complexes is a proper homotopy equivalence if and only if it induces isomorphisms of the fundamental groups at ∞ and of the locally ﬁnite cohomology groups. We shall only need the following special case :

(This is an abstract version of the mapping torus trick of M. Mather [91].) 65

66

Ends of complexes

If (D, f, g, h) is a domination of C and D is a free A-module chain complex then the left hand side is A-module chain equivalent to C, and the right hand side is a free A-module chain complex. The converse is trivial. (ii) A ﬁnite f.g. projective chain complex is ﬁnitely dominated since it is a direct summand of a ﬁnite f.g. free chain complex. See Ranicki [120] for the proof of the converse, including the construction from a ﬁnite domination (D, f, g, h) of an explicit f.g. projective chain complex P chain equivalent to a ﬁnitely dominated C. (In fact, by L¨ck and Ranicki [87] such P can u be constructed from the chain homotopy projection f g : D−→D.) The projective class of a ﬁnitely dominated A-module chain complex C is deﬁned by
∞

(This is another application of the algebraic Mather trick cited in the proof of 6.1 (i).) We refer to Milnor [99] and Cohen [30] for accounts of simple homotopy theory, and to Rosenberg [136] for algebraic K-theory. See Ranicki [120, 121, 123] for more detailed accounts of the algebraic theories of ﬁniteness obstruction and torsion. In the applications to topology A = Z[π] is a group ring. Here are some examples when the algebraic K-groups are known : Example 6.4 (i) The reduced projective class group of the group ring of the quaternion group Q(8) = {±1, ±i, ±j, ±k} is K0 (Z[Q(8)]) = Z2 , with generator [P ] the projective class of the f.g. projective Z[Q(8)]-module P = im(p) deﬁned by the image of the projection p = 1 − 8N −3N 21N 8N : Z[Q(8)] ⊕ Z[Q(8)] −→ Z[Q(8)] ⊕ Z[Q(8)] ,

68 with N =
g∈Q(8)

Ends of complexes g ∈ Z[Q(8)] such that N 2 = 8N (cf. Ranicki [129]).

(ii) The Whitehead group of the cyclic group of order 5 Z5 = {t | t5 } is W h(Z5 ) = Z , with generator the torsion τ (u) of the unit u = 1 − t + t2 ∈ Z[Z5 ]• . (iii) For many inﬁnite torsion-free groups π with ﬁnite classifying space Bπ K0 (Z[π]) = W h(π) = 0 , by the algebraic K-theory version of the integral Novikov conjecture. In particular, this is the case for the fundamental groups of hyperbolic manifolds (Farrell and Jones [50]). Thus tame ends of high dimensional hyperbolic manifolds have unique collarings. See Chapter D.3 of Benedetti and Petronio [6] for an account of the ends of hyperbolic manifolds, including a geometric proof that certain ends of hyperbolic manifolds can be collared. See also §12.6 of Ratcliﬀe [133]. The torsion of a homotopy equivalence f : K−→L of ﬁnite CW complexes is deﬁned by τ (f ) = τ (f : C(K)−→C(L)) ∈ W h(π1 (L)) . The homotopy equivalence f is simple if τ (f ) = 0, which is the case if and only if f is homotopic to the composite of a ﬁnite sequence of elementary expansions and collapses. A ﬁnite structure (Y, φ) on a space X is a ﬁnite CW complex Y together with a homotopy equivalence φ : X−→Y . A topological space is homotopy ﬁnite if it admits a ﬁnite structure, i.e. if it is homotopy equivalent to a ﬁnite CW complex. A simple homotopy type on a space X is an equivalence class of ﬁnite structures (Y, φ) on X, subject to the equivalence relation (Y, φ) ∼ (Y , φ ) if τ (φ φ−1 : Y −→Y ) = 0 ∈ W h(π1 (X)) . The simple homotopy types on a connected CW complex X are in one-toone correspondence with the simple chain homotopy types (if any) on the cellular Z[π1 (X)]-module chain complex C(X) of the universal cover X of X. Example 6.5 A compact AN R is homotopy ﬁnite (West [168]), and has a canonical simple homotopy type (Chapman [24]). For a ﬁnite CW complex this is the simple homotopy type determined by the cellular structure.

6. Projective class and torsion

69

An h-cobordism is a manifold cobordism (W ; M, M ) such that the inclusions M −→W , M −→W are homotopy equivalences, with torsion τ (W ; M, M ) = τ (M −→W ) ∈ W h(π1 (W )) . An s-cobordism is an h-cobordism (W ; M, M ) with τ (W ; M, M ) = 0 ∈ W h(π1 (W )) . The s-cobordism theorem is given by : Theorem 6.6 (Barden, Mazur, Stallings) An n-dimensional h-cobordism (W ; M, M ) is an s-cobordism if (and for n ≥ 6 only if ) (W ; M, M ) is homeomorphic rel M to M × (I; {0}, {1}). The original h-cobordism theorem of Smale is the special case π1 (W ) = {1}, when every h-cobordism is an s-cobordism, by virtue of W h({1}) = 0. Kervaire [83] is the standard account of the s-cobordism theorem. A domination (Y, f, g, h) of a space X by a space Y is deﬁned by maps f : X−→Y , g : Y −→X and a homotopy h : gf 1 : X−→X, so that X is a homotopy direct summand of Y . A topological space is ﬁnitely dominated if it is dominated by a ﬁnite CW complex. Proposition 6.7 (i) A topological space X is dominated by a CW complex if and only if it has the homotopy type of a CW complex. (ii) A topological space X is ﬁnitely dominated if and only if X × S 1 has the homotopy type of a ﬁnite CW complex. Proof For any maps f : X−→Y , g : Y −→X M. Mather [91] deﬁnes a homotopy equivalence T (gf ) T (f g) of the mapping tori (14.2) – see Chapter 14 below for the deﬁnition of the mapping torus. (i) If (Y, f, g, h) is a domination of a space X by a CW complex Y then X ×S 1 T (gf ) T (f g) determines a domination of X by the CW complex T (f g). It follows from the homotopy equivalences X X × R T (f g) that X is homotopy equivalent to a CW complex, namely the inﬁnite cyclic cover T (f g) of T (f g) (as deﬁned in Chapter 14 below). The converse is trivial. (ii) As for (i), noting that T (f g) is a ﬁnite CW complex for a ﬁnite CW complex Y . Let X be a regular cover of a CW complex X, with group of covering translations π. If X is ﬁnitely dominated then C(X) is a ﬁnitely dominated Z[π]-module chain complex, and the projective class of X with respect to X is deﬁned by [X] = [C(X)] ∈ K0 (Z[π])

The main results of ﬁniteness obstruction theory are summarized in : Theorem 6.8 (Wall [163]) (i) A connected CW complex X is ﬁnitely dominated (resp. homotopy ﬁnite) if and only if the fundamental group π1 (X) is ﬁnitely presented and the cellular chain complex C(X) of the universal cover X is a ﬁnitely dominated (resp. chain homotopy ﬁnite) Z[π1 (X)]module chain complex. (ii) The reduced projective class of a ﬁnitely dominated CW complex X with respect to the universal cover X is the ﬁniteness obstruction [X] ∈ K0 (Z[π1 (X)]) , such that [X] = 0 if and only if X is homotopy ﬁnite. (iii) If π is a ﬁnitely presented group and P is a f.g. projective Z[π]-module there exists a ﬁnitely dominated CW complex X with π1 (X) = π , [X] = [P ] ∈ K0 (Z[π]) . Idea of proof (i) It is clear that if X is ﬁnitely dominated (resp. homotopy ﬁnite) then π1 (X) is ﬁnitely presented and C(X) is ﬁnitely dominated (resp. chain homotopy ﬁnite), so only the converse has to be veriﬁed. The original proof in [163] was simpliﬁed by Hofer [67] using the algebraic theory of ﬁniteness obstruction of Ranicki [120], as follows. A connected CW complex X with ﬁnitely presented π1 (X) is homotopy equivalent to a CW complex (also denoted by X) with ﬁnite 2-skeleton. If D is a based free Z[π1 (X)]module chain complex with Dr = C(X)r for r = 0, 1, 2 and f : D−→C(X) is a chain equivalence which is the identity in dimensions ≤ 2 then the method of attaching cells to kill homotopy classes can be used to realize D by a CW complex Y with a homotopy equivalence f : Y −→X inducing f : C(Y ) = D−→C(X). Consider ﬁrst the case when C(X) is chain homotopy ﬁnite, so that D can be chosen to be a f.g. free Z[π1 (X)]-module chain complex, Y is ﬁnite and X is homotopy ﬁnite. In the other case C(X) is chain homotopy ﬁnitely dominated, and X × S 1 is such that the cellular Z[π][z, z −1 ]-module chain complex of the universal cover (X × S 1 ) = X × R C(X × R) = C(X) ⊗Z C(R) is chain homotopy ﬁnite, so that X × S 1 is homotopy ﬁnite (by the ﬁrst

We now recall the deﬁnitions of ‘forward tameness’ and ‘forward collaring’, and derive various consequences. For a forward tame σ-compact metric space W the homology at ∞ S ∞ (W ) of Chapter 3 is shown in 7.10 to be just the homology of e(W ),
∞ H∗ (W ) = H∗ (e(W )) ,

Deﬁnition 7.1 (Quinn [116]) Let W be a locally compact Hausdorﬀ space. (i) The space W is forward tame if there exists a closed cocompact subspace V ⊆ W such that the inclusion V × {0}−→W extends to a proper map q : V × [0, ∞)−→W . (ii) The space W is forward collared if there exists a closed cocompact AN R subspace V ⊆ W such that the identity V × {0}−→V extends to a proper map q : V × [0, ∞)−→V . Forward tameness is a homotopy theoretic version of Siebenmann’s compression axiom [148, 149]. Forward tameness will be interpreted as a homotopy pushout property in Chapter 12. In Parts Two and Three we shall be particularly concerned with forward tameness and collaring for the ends of inﬁnite cyclic covers of ﬁnite CW complexes. In Chapter 13 we shall give a homotopy theoretic criterion for forward tameness of such an end, and in Chapter 23 we shall give a homological criterion. The ‘locally ﬁnite projective class at ∞’ of a forward tame CW complex constructed in Chapter 10 is an algebraic K-theory obstruction to forward collaring. In Chapter 13 it will be shown that a forward tame end of an inﬁnite cyclic cover of a ﬁnite 75

(iii) Let η be a real n-plane vector bundle over a compact space K. The total space E(η) is forward collared, with E(η) K. The one-point compactiﬁcation E(η)∞ and the end space e(E(η)) are such that E(η)∞ = T (η) , e(E(η)) S(η)

(v) The mapping telescope Tel(fj ) of a direct system of maps fj : Xj −→ Xj+1 (2.3) is forward collared : the projection e(Tel(fj ))−→Tel(fj ) is a homotopy equivalence by 2.5, so that 7.2 (iii) applies with V = W = Tel(fj ). The one-point compactiﬁcation Tel(fj )∞ is contractible. (vi) If X is a compact subset of the interior of a compact manifold M and X has an I-regular neighbourhood in M in the sense of Siebenmann [148] then a result of Ferry and Pedersen [57, p. 487] can be used to show that M \X is forward tame. In particular, if X is 1-LCC embedded in M , and has the shape of a CW complex (for example, if X has the homotopy type of a CW complex) then M \X is forward tame by [148, p. 56]. Remark 7.4 (i) If a space W has ﬁnitely many ends, then W is forward tame (resp. forward collared) if and only if each end of W is forward tame (forward collared). (ii) In 11.14 below we shall show that an AN R space W is forward tame if and only if W × S 1 is inﬁnite simple homotopy equivalent to a forward collared AN R space X. Proposition 7.5 Let W be a forward tame space and let V ⊆ W be a closed cocompact subspace for which the inclusion u : V −→W extends to a proper map q : V × [0, ∞)−→W . Let (W , V ) be a cover of (W, V ) with group of covering translations π.

with respect to which W has a bounded CW complex structure. W is contractible, and hence ﬁnitely dominated. The end space e(W ) is not ﬁnitely dominated, since π0 (e(W )) is (countably) inﬁnite : no two of the proper paths ωn : [0, ∞) −→ W ; t −→ (n + t, n) (n ≥ 0) are proper homotopic. Moreover, since every closed cocompact subspace V ⊆ W has only ﬁnitely many path components, e(W ) is not dominated by any such V . By 7.5 (i), W is not forward tame. In fact, W ∞ is homotopy equivalent to the Hawaiian earring, the subspace of the plane consisting of circles of radius 1/n and centre (1/n, 0) (n = 1, 2, 3, . . .), which is well-known not to be homotopy equivalent to a CW complex. (This can be proved as follows. By deﬁnition, a space X is weakly locally contractible if every point x ∈ X has a neighbourhood U ⊆ X which is contractible in X. Any space homotopy equivalent to X is then also weakly locally contractible (Dugundji [38, p. 375, Exercise 7]). Every CW complex is (weakly) locally contractible. The Hawaiian earring is not weakly locally contractible at 0.) Remark 7.8 A locally compact space W is movable at the end if for each cocompact subspace U of W there exists a cocompact subspace V ⊆ U of W such that for each cocompact subspace Z of W there is a homotopy f : V × I−→U such that f0 = inclusion : V −→U and f (V × {1}) ⊆ Z (see Geoghegan [63]). Movability at the end is a precursor to forward tameness and originated in shape theory with the notion of movability for compacta due to Borsuk [7]. End movability has played a role in the theory of ends of open 3-manifolds (see Brin and Thickstun [11]). Clearly a forward tame space is movable at the end. The converse does not hold as the space W in 7.7 is movable at the end but not forward tame. Remark 7.9 (i) The CW complex W = {(x, n) ∈ R2 | x ≥ 0, n ∈ N} is a forward tame locally ﬁnite CW complex with W e(W ) not ﬁnitely dominated. (This example appears again in 12.6.) (ii) If W is forward tame and path-connected at ∞, then W is semistable at ∞ (2.23). Proposition 7.10 Let W be a σ-compact metric space W which is forward tame and path-connected at ∞, and let W ⊇ W0 ⊇ W1 ⊇ . . . be a sequence of closed cocompact subspaces such that Wj = ∅.
j

82

Ends of complexes

(i) W has stable π1 at ∞ (2.23), and the fundamental group of the end space e(W ) is the fundamental group at ∞ of W ,
∞ π1 (e(W )) = π1 (W ) .

The chain complex S(e(W )) is free, and S ∞ (W ) is chain equivalent to a free chain complex by 7.5 (ii) and 6.1 (i). Any homology equivalence of free chain complexes is a chain equivalence, so that α is a chain equivalence. (In fact, it is possible to deﬁne a chain homotopy inverse α−1 : S ∞ (W ) −→ S ∞ (V ) −→ S(V ) −→ S(e(W )) with u−1 a chain homotopy inverse to the inclusion u : S ∞ (V )−→S ∞ (W ), which is a chain equivalence by 3.13.) (iv) As for (ii), using 5.3. Proposition 7.11 (i) The one-point compactiﬁcation W ∞ of a forward tame AN R W is an AN R. (ii) The one-point compactiﬁcation W ∞ of a forward tame AN R W is such that there exists a pointed ﬁnite CW complex (X, x0 ) with a homotopy equivalence (W ∞ , ∞) (X, x0 ) .
u−1 q

84

Ends of complexes

Proof (i) Let V be a closed cocompact subset of W for which there is a proper map q : V × [0, ∞)−→W extending the inclusion q0 : V −→W . Suppose that W ∞ is a closed subset of some (separable) metric space X. Then W is closed in X\{∞}, so there exist a neighbourhood N of W in X\{∞} and a retraction r : N −→W . Let U1 ⊆ X be an open subset containing ∞ such that U 1 ∩ W \V = ∅ . Let U2 ⊆ X\{∞} be an open subset such that W ⊆ U2 ⊆ U 2 ⊆ N , (U 1 \r−1 int(V )) ∩ U 2 = {∞} . Let ρ : (U 1 ∪ U 2 )\{∞}−→[0, ∞] be a map such that ρ−1 (∞) = U 1 \(r− (int(V )) ∪ {∞}) , ρ−1 (0) = U 2 . The map r : U 1 ∪ U 2 −→W ∞ deﬁned by if x ∈ U 1 \r−1 (int(V )) , r(x) = q(r(x), ρ(x)) if x ∈ r−1 (V ) ,   r(x) if x ∈ U 2 \r−1 (V ) is a retraction. (ii) From (i) we know that W ∞ is a compact AN R. The result now follows from some well-known facts. The Triangulation Theorem of Chapman [23, p. 83] states that every Q-manifold M can be triangulated, i.e. is homeomorphic to K × Q for a polyhedron K. The AN R Theorem of Edwards ([23, p. 106]) states that if X is an AN R then X × Q is a Q-manifold. Applied to our context we have that there exists a ﬁnite CW complex X such that W ∞ × Q ∼ X × Q where Q is the Hilbert cube. If (x0 , q) ∈ X × Q cor= responds to (∞, 0) under such a homeomorphism, then (W ∞ , ∞) (X, x0 ). West [168] originally proved that compact AN R’s have the homotopy type of ﬁnite CW complexes. The argument above is a well-known alternative proof of West’s theorem (see Chapman [23]). The relevance of this argument is that it shows that pointed compact AN R’s have the homotopy type of pointed ﬁnite CW complexes. Example 7.12 Jacob’s ladder X (2.26) is an AN R whose one-point compactiﬁcation X ∞ is not locally contractible at {∞}. Thus X ∞ is not an AN R and 7.11 (i) implies that X is not forward tame. Similarly for the space W of 4.14 (which is proper homotopy equivalent to X – in 9.6 below forward tameness will be shown to be a proper homotopy invariant, in fact an invariant of the ‘proper homotopy type at ∞’). The one-point compactiﬁcation of a CW complex does not in general have the homotopy type of a CW complex.
 ∞ 

7. Forward tameness

85

Example 7.13 As in 3.18 let N = {0, 1, 2, . . .} have the discrete topology, the only topology compatible with CW status, so that
∞

is an embedding. N∞ does not have the homotopy type of a CW complex, since CW complexes are locally path-connected. Thus N is not forward tame, by 7.11 (ii). Proposition 7.14 Suppose W is a forward tame AN R written as
∞

We now formulate the deﬁnition of ‘reverse tameness’, which is a generalization of the manifold tameness of Siebenmann [140]. Deﬁnition 8.1 Let W be a locally compact Hausdorﬀ space. (i) The space W is reverse tame if for every cocompact subspace U ⊆ W there exists a cocompact subspace V ⊆ W with V ⊆ U such that U is dominated by U \V , by a homotopy h : W × I−→W such that : (a) h0 = idW , (b) ht |(W \U ) = inclusion : W \U −→W for every t ∈ I, (c) h(U × I) ⊆ U , (d) h1 (W ) ⊆ W \V . (ii) The space W is reverse collared if for every cocompact subspace U ⊆ W there exists a cocompact subspace V ⊆ U such that U \V is a strong deformation retract of U , in which case there exists a homotopy h : W × I−→W as in (i). By analogy with 7.2 and 7.3 : Proposition 8.2 A reverse collared space is reverse tame. In Parts Two and Three we shall be particularly concerned with reverse tameness and collaring for the ends of inﬁnite cyclic covers of ﬁnite CW complexes, and with the connections with forward tameness and collarings. In Chapter 13 we shall give a homotopy theoretic criterion for reverse tameness of such an end, and in Chapter 23 we shall give a homological criterion. The ‘projective class at ∞’ of a reverse tame CW complex constructed in Chapter 10 is an algebraic K-theory obstruction to reverse collaring. In Chapter 13 it will be shown that a reverse tame end of an inﬁnite cyclic 92

8. Reverse tameness

93

cover of a ﬁnite CW complex is reverse collared if and only if this invariant vanishes. Example 8.3 (i) Let (L, K ⊆ L) be a pair of compact spaces, and let W = L ∪K×{0} K × [0, ∞) . Then W is reverse collared : for each cocompact subspace U ⊂ W there exists t > 0 such that V = K × (t, ∞) ⊂ U is a cocompact subspace with U \V a deformation retract of U . (ii) Let (M, ∂M ) be a compact manifold with boundary. The boundary is collared (1.8), so that the interior int(M ) = M \∂M ∼ M ∪∂M ×{0} ∂M × [0, ∞) = is reverse collared by (i). Remark 8.4 (i) If a space W has ﬁnitely many ends, then W is reverse tame (resp. reverse collared) if and only if each end of W is reverse tame (resp. reverse collared). (ii) In a reverse tame space W every closed cocompact subspace U ⊆ W is dominated by a compact subspace, the closure of U \V in the terminology of 8.1. In particular, W is dominated by a compact subspace. (iii) In Chapter 13 below we shall show that a ﬁnitely dominated inﬁnite cyclic cover of a ﬁnite CW complex is reverse tame (as well as forward tame). Proposition 8.5 For an AN R space W the following are equivalent : (i) W is reverse collared, (ii) there exists a sequence of compact AN R subspaces
∞

The strong deformation retraction of Vj to Vj \Vj+1 extends to a strong deformation retraction of W to Kj . Hence, the inclusion Kj −→W is a homotopy equivalence. Since Kj is a retract of W , Kj is an AN R. (ii) =⇒ (i) Since the inclusion Kj −→W is a homotopy equivalence and W, Kj are AN R’s, Kj is a strong deformation retract of W for each j = 1, 2, 3, . . . . Given a cocompact subspace U ⊆ W , there exists j ≥ 1 such that W \U ⊆ Kj . Let V = W \Kj . A strong deformation retraction of W to Kj restricts to a strong deformation retraction of U to U \V . Example 8.6 Jacob’s ladder X (2.26) is an AN R with H1 (X) = Z[z] an inﬁnitely generated f.g. free Z-module. A compact AN R is ﬁnitely dominated (and in fact homotopy ﬁnite, West [168]), so that its homology consists of f.g. Z-modules. Thus X is not homotopy equivalent to a compact AN R and 8.5 implies that X is not reverse tame. Similarly for the space W of 4.14 (which is proper homotopy equivalent to X – in 9.8 below reverse tameness will be shown to be a proper homotopy invariant for AN R spaces such as X, W ). Proposition 8.7 Suppose W is a space with the property that every cocompact subspace X ⊆ W contains an AN R cocompact subspace Y ⊆ X which is closed in W . Then the following conditions are equivalent : (i) W is reverse tame, (ii) every closed AN R cocompact subspace X ⊆ W is ﬁnitely dominated, (iii) every cocompact subspace X ⊆ W contains a ﬁnitely dominated (AN R) cocompact subspace Y ⊆ X which is closed in W . Moreover, if W is also σ-compact, then the above conditions are equivalent to : (iv) there exists a sequence of ﬁnitely dominated (AN R) closed cocompact subspaces W = W0 ⊇ W1 ⊇ W2 ⊇ . . . with Wj = ∅.
j

Proof (i) =⇒ (ii) X is compactly dominated, by 8.4 (ii). Since X is an AN R (and hence homotopy equivalent to a CW complex), it follows that X is ﬁnitely dominated. (ii) =⇒ (iii) This follows immediately from the hypothesis. (iii) =⇒ (i) Let U ⊆ W be a cocompact subspace. By hypothesis there exists an AN R cocompact subspace X ⊆ U which is closed in W . Now there exists a ﬁnitely dominated cocompact subspace Y ⊆ X which is closed in W . We may assume that Y is disjoint from the frontier Fr(X) of X. Since Y is

(ii) =⇒ (iii), (iii) =⇒ (i) These follow from 8.7. Finally, it is clear that if W is countable, then (iii) and (iv) are equivalent. Corollary 8.10 A reverse tame strongly locally ﬁnite CW complex W is ﬁnitely dominated. Deﬁnition 8.11 A space W is reverse π1 -tame if it is reverse tame and each end has stable π1 at ∞ (2.23). Remark 8.12 An open manifold with compact boundary (W, ∂W ) and one end is tame in the sense of Siebenmann [140] if : (i) W is π1 -stable at ∞, so that there exists a sequence W ⊇ W0 ⊃ W1 ⊃ W2 ⊃ . . . of path-connected cocompact subspaces with cl(Wj ) = ∅
j

Proposition 9.15 Let W be an AN R which has arbitrarily small closed cocompact subsets which are AN R’s. The following conditions on W are equivalent : (i) W is both forward and reverse tame, (ii) W is forward tame and the end space e(W ) is ﬁnitely dominated, (iii) there exist a proper map W −→[0, ∞) and a space Y such that W is bounded homotopy equivalent at ∞ to the projection Y × [0, ∞)−→ [0, ∞). Moreover, if these conditions are satisﬁed Y is homotopy equivalent to e(W ). Proof (i) implies (ii) by 7.5 (i) and 8.7. (ii) implies (iii) by 9.14. (iii) implies (i) by 9.12 (i), (ii). If these conditions are satisﬁed Y e(W ) by 9.12 (iii). Theorem 9.16 Let W be an AN R which has arbitrarily small closed cocompact subsets which are AN R’s. The following conditions on W are equivalent : (i) W is both forward and reverse tame and e(W ) is homotopy equivalent to a ﬁnite CW complex, (ii) there exist a proper map W −→[0, ∞) and a ﬁnite CW complex K so that W is bounded homotopy equivalent at ∞ to the projection K × [0, ∞)−→[0, ∞), (iii) there exists a ﬁnite CW complex K so that W is proper homotopy equivalent at ∞ to the projection K × [0, ∞)−→[0, ∞). Moreover, if these conditions are satisﬁed K is homotopy equivalent to e(W ).

The metric space X is locally compact, forward tame and reverse tame. However, the end space e(X) has inﬁnitely many components, one for each 1 element of { 1 , 3 , 1 , . . . , 0}, so that it is not ﬁnitely dominated. Since X 2 4 is not locally connected, X is not an AN R and therefore this does not contradict 9.15. However, this example does contradict Quinn [116, p. 466]. Proposition 9.18 Let W be a forward tame AN R which has arbitrarily small closed cocompact subsets which are AN R’s (e.g. a strongly locally ﬁnite CW complex or a Hilbert cube manifold). The following conditions on W are equivalent : (i) W is reverse tame, (ii) W is reverse π1 -tame, (iii) the end space e(W ) is ﬁnitely dominated. Proof (i) =⇒ (ii) W has stable π1 at ∞ by 7.11. The other implications follow from 9.15.

10
Projective class at inﬁnity

We associate to a reverse π1 -tame space W the ‘projective class at ∞’ ∞ [W ]∞ ∈ K0 (Z[π1 (W )]) (10.1) with image the projective class (= Wall ﬁniteness obstruction) [W ] ∈ K0 (Z[π1 (W )]). The projective class at ∞ is an obstruction to reverse collaring W , which for an open manifold is the end obstruction of Siebenmann [140]. In 10.13 we prove a form of Poincar´ e duality (originally due to Quinn [116]) that a manifold end is forward tame if and only if it is reverse tame, subject to suitable fundamental group conditions, in which case the locally ﬁnite projective class at ∞ is the Poincar´ e dual of the projective class at ∞ (10.15). We associate to a forward tame CW complex W the ‘locally ﬁnite projective class’ [W ]lf ∈ K0 (Z[π1 (W )]) (10.4), and a ‘locally ﬁnite projective class at ∞’ [W ]lf ∈ K0 (Z[π1 (e(W ))]) (10.8), such that [W ]lf is the image of ∞ [W ]lf . The locally ﬁnite projective class at ∞ is an obstruction to forward ∞ collaring W . If W is both forward and reverse tame then e(W ) is ﬁnitely dominated, with ﬁniteness obstruction [e(W )] = [W ]∞ − [W ]lf ∈ K0 (Z[π]) ∞
∞ where π = π1 (e(W )) = π1 (W ).

with only a ﬁnite number of the coeﬃcients ai ∈ Z non-zero. Give R+ = [0, ∞) the CW structure with one 0-cell at each n ∈ N ⊂ R+ , and 1-cells [n, n+1], so that the cellular chain complex is the Z[z]-module chain complex 1−z −→ Z[z] , C(R+ ) : Z[z] −− with z acting by R+ −→R+ ; x−→x+1. Let Z[[z]] be the ring of formal power series
∞

is split injective. Since it suﬃces to prove that V is reverse tame, we assume that π = π1 (e(W ))−→π1 (W ) is split injective. All covers below are induced from the universal cover W −→W . By 9.18 it suﬃces to prove that e(W ) is ﬁnitely dominated. Since e(W ) has the homotopy type of a CW complex (7.6), it suﬃces to show that S(e(W )) is a ﬁnitely dominated Z[π]-module chain complex (6.9 (i)). According to 7.10 (iv) there is deﬁned a Z[π]-module chain equivalence S(e(W ))
C(S(W )−→S lf,π (W ))∗+1 .

Since S lf,π (W ) is ﬁnitely dominated by 7.5 (iii), it suﬃces to prove that S(W ) is a ﬁnitely dominated Z[π]-module chain complex. The Z[π]-module chain complex S(W ) is ﬁnitely dominated if and only if the n-dual Z[π]module chain complex S(W )n−∗ = HomZ[π] (S(W ), Z[π])n−∗ is ﬁnitely dominated. Now (W, ∂W ) is an open n-dimensional Poincar´ pair, so there is e a Z[π]-module chain equivalence S(W )n−∗ S lf,π (W , ∂ W ). Use 7.5 (iii) again to conclude that S lf,π (W ) is ﬁnitely dominated, from which it follows that S lf,π (W , ∂ W ) is ﬁnitely dominated. Conversely, suppose W is reverse π1 -tame. In order to apply 10.12, let W ⊇ W0 ⊇ W1 ⊇ . . . be as in 10.12. Since it suﬃces to prove that W0 is forward tame, we assume that W = W0 and all covers below are induced from

are given algebraically by the projections in the splittings of [118] Lq (Z[π][z, z −1 ]) = Lq (Z[π]) ⊕ Lr (Z[π]) . n n n−1 We conclude this chapter with the applications of inﬁnite simple homotopy theory to the detection of reverse and forward collaring. In 11.13 we prove that a locally ﬁnite CW complex W is inﬁnite simple homotopy equivalent to a reverse collared CW complex if and only if every coﬁnite subcomplex of W is homotopy equivalent to a ﬁnite CW complex. The proof will require the following technical result. Lemma 11.12 Let W be a strongly locally ﬁnite CW complex with a coﬁnite subcomplex V ⊆ W homotopy equivalent to a ﬁnite CW complex. Let A ⊆ W be a ﬁnite subcomplex with W \V ⊆ A. Then there exist a coﬁnite subcomplex U ⊆ V with A ∩ U = ∅, a ﬁnite subcomplex B ⊆ V with W \(A ∪ U ) ⊆ B, a ﬁnite CW complex B with B ∩ (A ∪ U ) = B ∩ (A ∪ U ) such that B, B are simple homotopy equivalent rel B ∩ (A ∪ U ), and a ﬁnite subcomplex K ⊆ W = A ∪ B ∪ U with A ⊆ K ⊆ A ∪ B such that the inclusion K−→W is a homotopy equivalence. Proof Let C = A ∩ V and let L be a ﬁnite CW complex homotopy equivalent to V . For some large n, we may assume that L is a subcomplex of

Thus f g| : W −→cyl(pW , k) is homotopic to the inclusion. Now use the fact that (W ∪ cyl(k)) × I ∪ (cyl(pW , k) × {0}) is a strong deformation retract of cyl(pW , k) × I in order to extend G+ . (ii) By 12.4 (ii) the identity M(pW , k)−→cyl(pW , k) is a homotopy equivalence rel {∞}. By (i) there is a homotopy equivalence g : cyl(pW , k)−→W ∞ rel {∞}. The composition M(pW , k)−→W ∞ is homotopic rel {∞} to the induced map. The following example shows that it is necessary to be careful about the topology on the mapping cylinders in Proposition 12.5.

for a pair (L, K ⊆ L) of compact spaces, so that W is forward collared, W L, W ∞ = L ∪ cK is the mapping cone of the inclusion K−→L, and e(W ) K. The homotopy pushout square of 12.5 is given up to homotopy equivalence by
K

w {∞}

L

u

(ii) As in 7.3 (ii) let (M, ∂M ) be a compact manifold with boundary, so that W = int(M ) = M \∂M is forward collared. The homotopy pushout square of 12.5 is given by
e(W ) ∂M

u w L ∪ cK

w {∞} u

W

u

M

wW

∞

= M/∂M

(iii) As in 7.3 (iii) let η be a real n-plane vector bundle over a compact space K, so that the total space W = E(η) is forward collared. The homotopy pushout square of 12.5 is given by
e(W ) S(η)

The non-compact spaces of greatest interest to us are equipped with a proper map to R. In this chapter we shall be particularly concerned with the inﬁnite cyclic cover W of a compact space W classiﬁed by a map c : W −→S 1 , which lifts to a proper map c : W = c∗ R−→R. We prove (!) that if W and W are connected and W is suﬃciently nice (such as a compact AN R) then W has two ends with (closed) neighbourhoods W such that W = W with W
+ + +

= c −1 [0, ∞) , W

−

= c −1 (−∞, 0] ⊂ W ∪W
−

∩W

−

= c −1 (0) compact.

−

W

−

W

+

∩W

−

W

+

+

W

The main result of this chapter is a geometric duality between forward and reverse tameness for the ends of an inﬁnite cyclic cover W of a compact + − AN R W : in 13.13 it is shown that W is forward tame if and only if W is reverse tame. In Chapter 15 we shall study ‘bands’ (W, c), which are compact spaces W with a map c : W −→S 1 such that the inﬁnite cyclic cover W = c∗ R of W is ﬁnitely dominated. It will be shown there that for an AN R band (W, c) the end spaces are such that e(W ) W W , e(W ) 147
+

In particular, manifold ribbons are geometric Poincar´ ribbons. e It is not required in 15.2 (iii) that d : X−→R be geometric Poincar´ e transverse at 0 ∈ R, so that X + ∩ X − need not be an (n − 1)-dimensional geometric Poincar´ complex (as is the case in 15.2 (ii)). e Manifolds with tame ends arise in the obstruction theory of Farrell [46, 47] and Siebenmann [145] for ﬁbring manifolds over S 1 , as ﬁnitely dominated inﬁnite cyclic covers of compact manifolds. These are particular cases of ‘bands’ :

15. Geometric ribbons and bands

175

Deﬁnition 15.3 (Siebenmann [143]) A band (W, c) is a compact space W with a map c : W −→S 1 such that the pullback inﬁnite cyclic cover W = c∗ R of W is ﬁnitely dominated, and such that the covering projection W −→W determines a bijection between the path components of W and those of W . Similarly for AN R band, CW band, geometric Poincar´ band and manie fold band. In Chapter 15 we shall be mainly concerned with bands (W, c : W −→S 1 ), particularly ones for which the inﬁnite cyclic cover (W , c : W −→R) is a ribbon. In 15.9 below it will be shown that if (W, c) is a AN R band with a π1 -fundamental domain (e.g. a manifold band) then (W , c) is an AN R ribbon. However, in general the inﬁnite cyclic cover (W , c) of a band (W, c) is not a ribbon, since the π1 -conditions of 15.1 may fail : Example 15.4 The inﬁnite cyclic cover (W , c) of the CW band (W, c) constructed in Example 13.16 for k = 1 is not a ribbon, since π1 (W ) = {1} + and π1 (Z) = {1} for any closed cocompact Z ⊆ W . In Chapter 19 we shall develop the ‘wrapping up’ construction of bands from ribbons, tying the two ends of a ribbon (X, d) to obtain a band (W, c) = (X, d) with an inﬁnite simple proper homotopy equivalence (W , c) (X, d). In particular, the inﬁnite cyclic cover of the band (S 1 , 1) is the ribbon 1 (S , 1) = (R, 1), and the wrapping up of the ribbon (R, 1) is the band (R, 1) = (S 1 , 1). Wrapping up will be used in Chapters 17–20 to prove that : (i) if W is a ﬁnite CW complex with a map c : W −→S 1 then (W, c) is a band (i.e. the inﬁnite cyclic cover W = c∗ R is ﬁnitely dominated) with a π1 -fundamental domain for the inﬁnite cyclic cover W = c∗ R of W if and only if (W , c : W −→R) is a ribbon, (ii) if X is an inﬁnite CW complex with a proper map d : X−→R then (X, d) is inﬁnite simple homotopy equivalent to a CW ribbon if and only if (X, d) is inﬁnite simple homotopy equivalent to the inﬁnite cyclic cover (W , c) of a CW band (W, c), if and only if d : X−→R is proper homotopic to a bounded ﬁbration, in which case X + , X − are both forward and reverse lf,π tame, and H∗ (X) = H∗−1 (X), (iii) if (X, d) is an n-dimensional geometric Poincar´ ribbon then X is a e ﬁnitely dominated (n − 1)-dimensional geometric Poincar´ complex, e (iv) if X is an open manifold of dimension n ≥ 5 with a proper map d : X−→R then (X, d) is a ribbon if and only if (X, d) is homeomorphic to the inﬁnite cyclic cover (W , c) of an n-dimensional manifold band (W, c), if

Remark 15.7 (i) If (W, c) is a connected AN R band, then W = c∗ R has + − exactly two ends W , W which will be shown in 15.9 to be both forward and reverse tame, with e(W ) The one-point compactiﬁcation W suspension of W + = W ∪ {pt.}
+

with W any regular cover of W with group of covering translations π. (ii) Let W be a connected ﬁnite CW complex with a map c : W −→S 1 such that the inﬁnite cyclic cover W = c∗ R of W is connected. By 13.13 + − W is forward tame if and only if W is reverse tame, with π1 (W ) = π1 (W ) = π1 (W ) = π . In 15.10 it will be proved that (W, c) is a band if and only if W is both + forward and reverse tame. In Chapter 23 it will be proved that W is reverse (resp. forward) tame if and only if the cellular Z[π]-module chain complex C(W + ) (resp. the π-locally ﬁnite cellular Z[π]-module chain complex C lf,π (W + )) is ﬁnitely dominated, with W + the universal cover of W + . It follows that if W is a ﬁnite n-dimensional geometric Poincar´ complex e + + the end W is forward tame if and only if W is reverse tame, if and only if (W, c) is a band, in which case W is a ﬁnitely dominated (n−1)-dimensional geometric Poincar´ complex and cap product with the fundamental class e
lf [W ] ∈ Hn (W ) = Hn−1 (W ) + + −

(iii) Let (V, ∂V ) be an open n-dimensional manifold with a compact boundary ∂V and one end which is both reverse and forward tame, so that (V ; ∂V, e(V )) is a ﬁnitely dominated n-dimensional geometric Poincar´ e cobordism. In Chapter 17 it will be shown that for n ≥ 5 there exists an open neighbourhood of the end W ⊂ V which is the inﬁnite cyclic cover of a compact n-dimensional manifold band (W, c) (the ‘wrapping up’ of V ) such that there exists a compact (n + 1)-dimensional cobordism (M ; ∂V × S 1 , W ) with homotopy equivalences (V ; ∂V, e(V )) × S 1 (M ; ∂V × S 1 , W ) , e(V ) W

is cocompact, and such that there exists a proper homotopy V × [0, 1)−→W + − extending the inclusion, so that W is forward tame, and hence W , W are + − forward tame. The spaces W , W are reverse tame by 13.13, and have stable π1 at ∞ by 7.11, so that they are reverse π1 -tame. (ii) The composite pW + + + p : e(W ) −− −→ W −→ W is a homotopy equivalence by 13.13. Reversing the role of the two ends of − − W similarly gives that the composite e(W )−→W −→W is a homotopy equivalence. (iii) The homology identities are given as in 13.15 (ii). + − (iv) These identities follow from (i), (ii) and W = W ∪ W .

are such that there is deﬁned a rel U homotopy ζ N ht ζ −N so that
N+ j=0
+ +

: gf
+

1 : W

+

−→ W

+

,

ζ j V dominates W

rel U .
+ −

(ii) As for (i), with the role of W , W

reversed.

Deﬁnition 15.14 Let (W, c) be a band. (i) The band (W, c) is positively relaxed if there exists a fundamental + domain (V ; U, ζU ) for W such that V dominates W rel U . (ii) The band (W, c) is negatively relaxed if there exists a fundamental

is injective and the cokernel is a f.g. projective A-module, or equivalently if ω becomes invertible over the Novikov rings A((z)) and A((z −1 )) (cf. 23.1 and 23.2 below). Let Ω be the set of Fredholm matrices in A[z, z −1 ], and let Λ = Ω−1 A[z, z −1 ] be the (noncommutative) localization of A[z, z −1 ] inverting Ω. This type of localization is a generalization of the single-element inversion of 2.28 (i). The injection A[z, z −1 ]−→Λ is a ring morphism with the universal property that a ﬁnite f.g. free A[z, z −1 ]-module chain complex C is A-ﬁnitely dominated if and only if H∗ (Λ ⊗A[z,z −1 ] C) = 0, by Ranicki

In Chapter 19 we shall develop a CW analogue of the Siebenmann twist glueing : given a connected CW ribbon (X, d) and a homotopy equivalence h : (X, d)−→(X, d) we use h to tie the two ends of X together to obtain a CW complex X(h) with a map d(h) : X(h)−→S 1 and with homotopy equivalences (X, d) (X(h), d(h)), X(h) T (h). In the special case when h : (X, d)−→(X, d) is a proper homotopy equivalence with h a covering translation or the identity we obtain a relaxed CW band (X[h], d[h]) with a proper homotopy equivalence (X, d) (X[h], d[h]) which is simple in the sense of the inﬁnite simple homotopy theory of Siebenmann [144]. In Chapter 20 the ‘wrapping up’ of a CW ribbon (X, d) (X, d) = (X[1], d[1]) will be used to prove that d : X−→R is proper homotopic to a proper bounded ﬁbration. The ‘relaxation’ of a CW π1 -band (W, c) is deﬁned in Chapter 20 to be the relaxed CW band (W , c ) = (W [ζ], c[ζ]) in the homotopy type of (W, c). In Chapters 24, 25 below we shall apply the homotopy theoretic twist glueing to study the ﬁbring obstructions of relaxed CW bands (which are distinguished by the property that the Nilcomponents vanish).

16
Approximate ﬁbrations

We characterize forward and reverse tameness for open manifolds in terms of approximate lifting properties; the main result (16.13) is that an open manifold W of dimension n ≥ 5 is forward tame and reverse tame if and only if there exists an open cocompact X ⊆ W with a manifold approximate ﬁbration d : X−→R. In Chapter 17 it will then be shown that an open manifold X of dimension n ≥ 5 is the total space of a manifold approximate ﬁbration d : X−→R if and only if it is the inﬁnite cyclic cover X = M of an n-dimensional manifold band (M, c). Deﬁnition 16.1 Let B be a metric space. (i) Let > 0, and let X, Y be spaces equipped with maps p : X−→B, q : Y −→B. A map f : X−→Y is an -homotopy equivalence over B if there exists a map g : Y −→X together with homotopies h : gf 1 : X −→ X , k : f g 1 : Y −→ Y

Corollary 16.4 If p : X−→R is a proper bounded ﬁbration such that p−1 [0, ∞) and p−1 (−∞, 0] are AN R’s, then X is forward tame and reverse tame. Proof Apply 16.3 and 9.12. The following result says that the existence of a bounded ﬁbration to R is a proper homotopy invariant. Proposition 16.5 If X is a metric space, f : X−→Y is a proper homotopy equivalence and p : Y −→R is a bounded ﬁbration then pf : X−→R is properly homotopic to a bounded ﬁbration d : X−→R. Proof Let g : Y −→X be a proper homotopy inverse for f and let G : idX gf be a proper homotopy. Let n0 = 0 and use the properness of p, f, g, and G to inductively choose ni ≥ i (i = 0, 1, 2, . . .) so that :

Let γ : R−→R be the P L homeomorphism such that γ(±ni ) = ±i (i = 0, 1, 2, . . .) . Then pf is properly homotopic to d = γpf : X−→R. One may check that γp is 3-close to γpf g and that γpf G : X × I−→R is a 3-homotopy. In order to show that d is a bounded ﬁbration, consider a lifting problem h : Z−→X and H : Z × I−→R such that dh(z) = H(z, 0) (z ∈ Z) . It follows from standard arguments that we may assume that Z is paracompact (see the proof of 16.3). This lifting problem induces a lifting problem f h : Z−→Y and H : Z × I−→R for γp : Y −→R. Suppose that p : Y −→R is a b-ﬁbration. Since γ is distance nonincreasing, γp is also a b-ﬁbration. There is a solution H : Z × I−→Y such that f h(z) = H(z, 0) (z ∈ Z) and γpH is b-close to H. Use the paracompactness of Z to deﬁne a map φ : Z−→(0, 1] such that H(z, 0) is 1-close to H(z, t) if t ≤ φ(z). Deﬁne H : Z × I −→ X ; (z, t) −→ Then h(z) = H(z, 0) (z ∈ Z) and dH is max{4, 3 + b}-close to H. Hence d is a bounded ﬁbration. Note that the proof above just requires X to be properly dominated by a bounded ﬁbration, rather than proper homotopy equivalent to one. We shall be mainly concerned with bounded ﬁbrations p : X−→B with p a proper map. Deﬁnition 16.6 (i) An approximate ﬁbration p : X−→B is a map which is an -ﬁbration for every > 0. (ii) A manifold approximate ﬁbration p : W −→B is an approximate ﬁbration such that W and B are manifolds (either ﬁnite dimensional without boundary or Hilbert cube manifolds), and such that p is a proper map. G(h(z), t/φ(z)) if 0 ≤ t ≤ 1/2 , g H(z, t/φ(z) − 1) if 1/2 ≤ t ≤ 1 .

16. Approximate ﬁbrations

191

Proposition 16.7 A map p : X−→R from a metric space X is an approximate ﬁbration if and only if for every > 0 it is -equivalent over R to the projection Y × R−→R for some space Y . Proof The proof of 16.3 shows that a space which is -equivalent over R to a product with R is a 3 -ﬁbration. And conversely, an -ﬁbration is 2 -equivalent over R to a product with R. More general versions of 16.3 and 16.7 are given in Hughes, Taylor and Williams [77, p. 47]. Proposition 16.8 Let X be a metric space with a map p : X−→[0, ∞) which is -homotopy equivalent at ∞ to the projection Y × [0, ∞)−→[0, ∞) for some space Y and some > 0. Then there exists u > 0 such that p is an 3 -ﬁbration over (u, ∞). Proof Let (f, g, X , Y ) : X−→Y × [0, ∞) be an -equivalence at ∞ with X = p−1 ([s, ∞)) and Y = Y × [t, ∞). For u much larger than s, t, the proof of 16.3 shows that p is a 3 -ﬁbration over (u, ∞). We shall use the sucking principle of Chapman [25, 26] to gain the control necessary to pass from bounded ﬁbrations to approximate ﬁbrations. This says that, for manifolds, there is essentially no diﬀerence between proper approximate ﬁbrations and proper -ﬁbrations for suﬃciently small > 0. At its simplest, the sucking principle takes on the following form. Theorem 16.9 (Chapman) For every n ≥ 5 and > 0 there exists δ > 0 such that if W is an open n-dimensional manifold or a Hilbert cube manifold and p : W −→R is a proper δ-ﬁbration, then p is -homotopic to a manifold approximate ﬁbration p : W −→R. The proof of 16.9 uses controlled engulﬁng. Corollary 16.10 If W is an open manifold of dimension ≥ 5 or a Hilbert cube manifold and p : W −→R is a proper bounded ﬁbration, then p is boundedly homotopic to a manifold approximate ﬁbration. Proof Suppose p is an -ﬁbration for some > 0. Choose δ > 0 by Theorem 16.9 so that any proper δ-ﬁbration from an n-manifold to R is 1-homotopic to a manifold approximate ﬁbration. Choose L > 0 so large that /L < δ and deﬁne γ : R −→ R ; x −→ x/L . Then γp : M −→R is an ( /L)-ﬁbration and is 1-homotopic to a manifold approximate ﬁbration p : M −→R. It follows that p is L-homotopic to γ −1 p which is a manifold approximate ﬁbration.

192

Ends of complexes

Proposition 16.11 For every n ≥ 5 there exists > 0 such that if (W, ∂W ) is an open n-dimensional manifold with compact boundary or a Hilbert cube manifold, and p : W −→[0, ∞) is a proper -ﬁbration over (a, ∞) for some a > 0, then p is properly homotopic to a map p such that p | : (p )−1 (a + 1, ∞) −→ (a + 1, ∞) is a manifold approximation ﬁbration. Proof The proof follows from Chapman’s proof of 16.9 in [25, 26]. The next example shows that this version of Chapman’s sucking principle fails for AN R’s. Example 16.12 We construct a non-manifold 2-dimensional CW band (W, c) such that c : W −→R is not properly homotopic to an approximate ﬁbration even though c is a proper bounded ﬁbration. Let D be the topologist’s dunce cap, the space obtained from the standard 2-simplex σ by identifying its three edges, two with the same orientation and one with the opposite orientation. So D is a CW complex with one 0-cell, one 1-cell, and one 2-cell. Let x be the 0-cell and let y be a point in the interior of the 2-cell. Let W be the space obtained from D by identifying x and y. Clearly, W can be given the structure of a ﬁnite 2-dimensional CW complex. (If D is obtained from the standard 2-simplex σ by identifying edges, and y is the barycentre of σ, the subdivision of σ obtained by starring from y induces a CW structure on D which in turn induces a CW structure on W . This CW structure on W has one 0-cell, four 1-cells, and three 2-cells.) Fix a classifying map c : W −→S 1 such that c−1 (1) = {x = y}. A Zequivariant lift c : W −→R is a bounded ﬁbration. This is because there is a strong deformation retraction of D to an arc in D joining x and y. This strong deformation retraction induces a bounded strong deformation retraction of W to a copy L of R such that c|L : L−→R is a homeomorphism. It also follows from 17.14 below that c is a bounded ﬁbration. Write
∞

W =
i=−∞

Di

where Di is a fundamental domain homeomorphic to D with vertices zi ∈ Di such that c(zi ) = i and Di ∩ Di+1 = {zi+1 }. Assume by way of contradiction that c is properly homotopic to an approximate ﬁbration d : W −→R. Since W and R are contractible, so is the homotopy ﬁbre of d. This in turn implies that d is a cell-like map (Ferry [53, p. 337]). It can also be veriﬁed directly that d is a cell-like map. Since d is proper, there exists an integer i such that d(zi ) < d(zi+1 ). Choose t such that d(zi ) < t < d(zi+1 ).

of Di , there exists v ∈ Di such that d(v) = t. Since zi ∈ d−1 (t), d−1 (t) has / at least two components : one containing u, and another containing v. This contradicts the cell-likeness of d−1 (t). Identify Di with D and assume i = 0. Thus, we have a map d : D−→I such that d(x) = 0, d(y) = 1, and d−1 (t) is cell-like for each t ∈ (0, 1). It follows that d−1 ( 1 ) is a cell-like subset of D which separates x from y. The 2 following claim provides a contradiction. Claim 2 If Z ⊆ D is a cell-like set and x ∈ Z, then Z does not separate D. / Proof Let π : σ−→D denote the quotient map. The proof is based on the following ﬁve items :

(i) Every proper 2-dimensional subpolyhedron P of D contains a free 1-dimensional face. For let Q be the subpolyhedron of σ which is the union of all 2-simplexes in π −1 (P ). It suﬃces to observe that that Q has a free 1-dimensional face not in ∂σ. (ii) Every subpolyhedron P of D is aspherical. If P = D this is true. On the other hand if P is proper, then (i) implies that P collapses to a 1-dimensional subpolyhedron which of course is aspherical. It also follows from Papakyriakopoulos [107, p. 19] that any subpolyhedron of D is aspherical. (iii) If P ⊆ Q ⊆ D\{x} are subpolyhedra, P contracts to a point in Q and Q contracts to a point in D\{x}, then there exists a contractible subpolyhedron P of D such that P ⊆ P ⊆ Q. By (ii) it suﬃces to ﬁnd a simply-connected subpolyhedron P with P ⊆ P ⊆ Q. This would follow from standard plane topology if Q ⊆ D\π(∂σ) (by adding appropriate complementary domains of P ). In the more general case, pass to the universal cover U of D\V where V is the interior of a small regular neighbourhood of x. Lift P and Q to subpolyhedra P , Q of U . The fundamental domain F for U is homeomorphic to a 2-cell so plane topology can be used in U to build P . The idea is to work inductively starting at the outermost translates of F which meet P and add appropriate complementary domains in that translate to P . (iv) If P ⊆ Q ⊆ D are subpolyhedra such that Q collapses to P across 2-simplexes and Q separates D, then P separates D. This can be veriﬁed by pulling P and Q back to σ and examining the several possible cases.

194

Ends of complexes

(v) No contractible 1-dimensional subpolyhedron P of D separates D. Since P collapses to a point and D is not separated by a point, a 1-dimensional analogue of (iv) is needed. As in (iv) this is veriﬁed by pulling back to σ and examining the several possible cases. Given these ﬁve items we can ﬁnish the proof of Claim 5. It follows from (iii) that we can write
∞

Z =
i=1

Zi

where Zi+1 ⊆ Zi and Zi is a contractible proper subpolyhedron of D. If Z separates D, then we may assume that each Zi separates D. It follows from (i) and (iv) that Z1 collapses to a 1-dimensional subpolyhedron P which does not separate D. Since Z1 is contractible, so is P , which is a contradiction to (v). Theorem 16.13 Let W be an open manifold of dimension ≥ 5 with compact boundary or a Hilbert cube manifold. The following conditions on W are equivalent : (i) W is both forward and reverse tame, (ii) W is forward tame and the end space e(W ) is ﬁnitely dominated, (iii) there exist an open cocompact X ⊆ W and a manifold approximate ﬁbration X−→R. Proof (i) ⇐⇒ (ii) follows from 9.15. (i) =⇒ (iii) follows from 9.14, 16.8 and 16.11. (iii) =⇒ (i) follows from 16.7 and 9.12. Theorem 16.13 is the existence part of the Teardrop Structure Theorem of Hughes, Taylor, Weinberger and Williams [76] in the simplest case (two strata, the lower stratum being a point). The next example shows that one cannot hope for a true analogue of 16.13 for AN R’s, even if one only wants a proper bounded ﬁbration. Example 16.14 For the CW complex W of 16.12 the result of adding a point at −∞ (thereby compactifying W − ) is an AN R X = W ∪ {−∞} which satisﬁes all the hypotheses of 16.13, yet no open cocompact subset of X admits a proper approximate ﬁbration to R.

Corollary 16.17 A manifold approximate ﬁbration d : W −→R is a ribbon. Proof The transversality conditions (i) in the deﬁnition of a ribbon (15.1) are given by manifold transversality. The homotopy conditions (ii) are given by 16.16. The homology conditions (iii) follow from the forward tameness of W (16.4) and 13.15 (ii). In Chapter 17 we shall prove that a manifold approximate ﬁbration d : W −→R of dimension ≥ 6 is proper homotopic to a lift c : M −→R of the classifying map c : M −→S 1 of a manifold band (M, c) such that W = M .

17
Geometric wrapping up

Wrapping up is a geometric compactiﬁcation procedure which for n ≥ 5 associates to an n-dimensional manifold approximate ﬁbration (X, d : X−→R) a relaxed n-dimensional manifold band (M, c) = (X, d : X−→S 1 ) with inﬁnite cyclic cover M = X, such that c : X−→R is an approximate ﬁbration properly homotopic to d, with X × S 1 homeomorphic to M × R. By 16.13 an open n-dimensional manifold W which is both forward and reverse tame has an open cocompact X ⊆ W with a manifold approximate ﬁbration d : X−→R, and the wrapping up provides a canonical collaring of the open (n + 1)-dimensional manifold W × S 1 with boundary M = X (17.10). We shall use wrapping up to prove that an AN R space X admits a proper bounded ﬁbration d : X−→R if and only if it is inﬁnite simple homotopy equivalent to the inﬁnite cyclic cover M of a CW band (M, c) (17.16). We also prove that an open manifold X admits a manifold approximate ﬁbration d : X−→R if and only if X is the inﬁnite cyclic cover M of a manifold band (M, c) (17.18). We begin in 17.1 with the wrapping up construction of a manifold band from a manifold approximate ﬁbration over R. Then in 17.11 we give some elementary consequences of the sucking principle (16.9). After observing in 17.12 that total spaces of proper bounded ﬁbrations over R are ﬁnitely dominated, we present the main characterizations (17.16 and 17.18). The next result of this chapter concerns bands. We know that ﬁnitely dominated inﬁnite cyclic covers of AN R bands admit proper bounded ﬁbrations to R (17.14), but might not admit any proper approximate ﬁbration to R (16.12, 16.14). However, in 17.20 we show that if the AN R band is allowed to vary up to simple homotopy type, then it will have a ﬁnitely dominated inﬁnite cyclic cover which admits a proper approximate ﬁbration to R. 197

ﬁbres, with ﬁbre X and monodromy ζ. (This fact is well-known to the experts, cf. Ferry and Pedersen [57, p. 492].) The plan of the proof of 17.1 is as follows. The manifold X is constructed using a variation of Chapman’s wrapping up construction (which in turn is a variation of Siebenmann twist glueing (15.17); see Hughes and Prassidis [75] for the precise relation between the two constructions). The input needed is the approximate isotopy covering property for manifold approximate ﬁbrations due to Hughes, allowing the standard shift map R−→R; s−→s + 1 to be lifted to a covering translation ζ : X−→X isotopic to the identity. Note that 17.1 (iv) is a direct consequence of (ii) and (iii). In dealing with isotopies G : X × I −→ X ; (x, s) −→ Gs (x) we always assume that G0 = id. : X−→X. Theorem 17.4 (Approximate Isotopy Covering) Let p : M −→B be a manifold approximate ﬁbration where M is a manifold without boundary of dimension n ≥ 5 or a Hilbert cube manifold (n = ∞). Let α be an open cover of B, and let g : B × I−→B be an isotopy. Then there exists an isotopy G : M × I−→M such that pGt is α-close to gt p for each t ∈ I. Comments on Proof. Note that ‘Approximate Isotopy Covering’ theorems are not well documented in the literature. However, the ‘Controlled Isotopy Covering Theorem’ in Hughes, Taylor and Williams [79] is derived from ‘Controlled Straightening’ in Hughes, Taylor and Williams [77], which in turn is derived from ‘Approximate Straightening’. Now ‘Approximate Straightening’ is just the ‘Approximation Theorem’ of Hughes [72]. The point of all of this is that the Approximate Isotopy Covering Theorem follows easily from the Approximation Theorem of [72]. Proposition 17.5 Let a1 , a2 , a3 > 0 be real numbers such that a3 > a1 + a2 . Let d : X−→R be a proper map, and let ζ : X−→X be a homeomorphism with dζ a2 -close to d. Also, let g : R −→ R ; x −→ x + a3 and let G : X−→X be a homeomorphism such that dG is a1 -close to gd. Then ζG : X −→ X is a covering translation of an inﬁnite cyclic cover of X/ζG, with U = d−1 (0) , V = ζGd−1 (−∞, 0] ∩ d−1 [0, ∞)

202

Ends of complexes

such that (V ; U, ζGU ) is a fundamental domain. Proof We shall show by induction on n ∈ Z+ that for each x ∈ X d(ζG)n x > dx + n(a3 − a1 − a2 ) . A similar argument would show d(ζG)−n x < dx − n(a3 − a1 − a2 ) and from these two inequalities it follows immediately that the orbit {(ζG)n x | n ∈ Z} is closed and discrete in X (the fact that d is proper must be invoked here), so that the action
Z × X −→ X ; (n, x) −→ (ζG)n x

R ∼ T (G1 ). On the other hand, since G1 is isotopic to idX there is a =

We now return to the proof of 17.1. It follows from 14.8 (i) that X ×

homeomorphism T (G1 ) ∼ X × S 1 . This completes the proof of 17.1. =

Remark 17.9 The wrapping up construction above is very similar to Chapman’s original construction [25, 26]. However, our construction is technically easier and more conceptual because we make use of the Approximate Isotopy Covering Theorem 17.4. Chapman only had the technology to approximately cover compactly supported isotopies and the shift map g1 : R−→R is far from compactly supported. Chapman had to truncate the shift map to get a compactly supported isotopy which he then approximately covered by a compactly supported isotopy on X. The upshot is that he constructed the fundamental domain V above, but the inﬁnite cyclic cover X of X was given as a proper open subset of X rather than equal to X. Using Chapman’s approach it is far from obvious that the generating covering translation ζ is isotopic to the identity, but it is immediate in our approach. Theorem 17.10 Let (W, ∂W ) be an open n-dimensional manifold with compact boundary and one end, with n ≥ 5 or n = ∞ (= Hilbert cube manifold). If W is forward tame and reverse tame then the end space e(W ) is homotopy equivalent to an open cocompact submanifold X ⊂ W with a proper map d : X−→R such that (X, d) = (M , c) is the ﬁnitely dominated inﬁnite cyclic cover of a relaxed n-dimensional manifold band (M, c) = (X, d) with X × S 1 homeomorphic to M × R, and with a rel ∂ homeomorphism (W, ∂W ) × S 1 ∼ (N \M, ∂W × S 1 ) = for a compact (n + 1)-dimensional manifold cobordism (N ; ∂W × S 1 , M ). Proof Combine 16.13 and 17.1. Let Q denote the Hilbert cube. Edwards proved that X × Q is a Hilbert cube manifold for any AN R X. (Recall our global assumption at the beginning of Chapter 1 that only locally compact, separable AN R’s are to be considered.) Proposition 17.11 (i) For any AN R X, the following are equivalent : (a) there exists a proper bounded ﬁbration d : X−→R , (b) for every > 0 there exists a proper -ﬁbration d : X−→R , (c) there exists a proper approximate ﬁbration d : X × Q−→R .

Proof The ‘if’ statement follows from 17.12. Let ζ : W −→W denote the (+1)-generating covering translation. Let Ks be given by 17.13. It remains to show that c has the (N + 1)-homotopy lifting property. This estimate arises because cp1 K1 : W × R−→R is (N +1)-close to p 2 : W × R−→R where p1 : W × R−→W and p 2 : W × R−→R are the projections (in fact, they are a distance at most N apart). Deﬁne g : W −→ W × R ; x −→ (x, c(x)) . Thus g is the natural embedding of W onto the graph Γ(c) of c. A lifting problem for c, say a homotopy F : Z × I−→R with an initial lift f : Z−→W so that F0 = cf , induces a lifting problem for p 2 with homotopy F but with initial lift given by gf : Z−→W × R. Of course, p 2 is a ﬁbration, so let F : Z × I−→W × R be a solution of this second problem so that F0 = gf . It follows that F = p1 K1 F is an (N + 1)-solution of the ﬁrst problem. Corollary 17.15 Let X be an open manifold of dimension n ≥ 5 or a Hilbert cube manifold. If X is an inﬁnite cyclic cover of a compact space, then there exists a manifold approximate ﬁbration d : X−→R. Proof Apply 16.10 and 17.14. Proposition 17.16 (i) For an AN R X the following are equivalent : (a) there exists a proper bounded ﬁbration d : X−→R , (b) for every > 0 there exists a proper -ﬁbration d : X−→R , (c) there exists a manifold approximate ﬁbration d : X × Q−→R , (d) X is ﬁnitely dominated and X × Q is an inﬁnite cyclic cover of a compact space, (e) X is inﬁnite simple homotopy equivalent to the ﬁnitely dominated inﬁnite cyclic cover W of a CW band (W, c) , (f) X is proper homotopy equivalent to the ﬁnitely dominated inﬁnite cyclic cover W of a CW band (W, c). (ii) For an open manifold X of dimension n ≥ 5 or a Hilbert cube manifold, the conditions of (i) are equivalent to : (g) there exists a manifold approximated ﬁbration d : X−→R , (h) X is ﬁnitely dominated and is an inﬁnite cyclic cover of a compact space. Proof (i) (a) ⇐⇒ (b) ⇐⇒ (c) by 17.11 (i). (c) =⇒ (d) because X × Q is a Hilbert cube manifold, so 17.1 implies that X × Q is the ﬁnitely dominated inﬁnite cyclic cover of a Hilbert cube

17. Geometric wrapping up

209

manifold band. (d) =⇒ (e) Let K be a compact space with inﬁnite cyclic cover K = X×Q. Since K is a compact Hilbert cube manifold, X is homeomorphic to Y × Q for some ﬁnite CW complex Y (by Chapman’s Triangulation Theorem). Then Y is a CW band with ﬁnitely dominated inﬁnite cyclic cover Y such that X × Q and Y × Q are homeomorphic. By the work of Chapman, this is what it means for X and Y to be inﬁnite simple homotopy equivalent. (e) =⇒ (f) is obvious. (f) =⇒ (a) by 16.5 and 17.14. (ii) (a) ⇐⇒ (g) by 17.11 (ii). (g) =⇒ (h) by 17.1. (h) =⇒ (a) by 17.14. Remark 17.17 (i) If the conditions of 17.16 (i) are satisﬁed, then the compact space of which X is an inﬁnite cyclic cover is a compact AN R and, hence, of the homotopy type of a ﬁnite CW complex (by the theorem of West [168]). (ii) It follows from 15.9 that the conditions of 17.16 (i) imply that X is proper homotopy equivalent to an AN R ribbon. For a CW ribbon (X, d) the converse is established in 20.3 (ii) : X is proper homotopy equivalent to the ﬁnitely dominated inﬁnite cyclic cover of a CW band. Corollary 17.18 An open manifold X of dimension ≥ 5 is the total space of a manifold approximate ﬁbration X−→R if and only if it is the inﬁnite cyclic cover X = M of a compact manifold band (M, c). Theorem 17.19 (i) For a strongly locally ﬁnite CW complex X with a ﬁnite number of ends, the following are equivalent : (a) X is forward and reverse tame, + (b) there exists a CW band (W, c) such that X and W are proper homotopy equivalent at ∞. (ii) For a manifold X with a ﬁnite number of ends of dimension ≥ 5 with compact boundary or a Hilbert cube manifold, the conditions above are equivalent to : (c) there exists a manifold band (W, c) such that W is homeomorphic to a closed cocompact subspace of X , (d) there exists a manifold band (W, c) such that W is homeomorphic to an open cocompact subspace of X.
+

210

Ends of complexes

Proof (i) (a) =⇒ (b) By 16.13 and 17.1 X × Q has an open cocompact subspace U which is the ﬁnitely dominated inﬁnite cyclic cover of a Hilbert cube manifold band (U, c1 ). Since U is homeomorphic to W × Q for some ﬁnite CW complex W , (W, c = c1 ◦ inclusion) is a CW band such that X + and W are proper homotopy equivalent at ∞. (b) =⇒ (a) by 15.9 (i), 9.6 and 9.8. (ii) (a) =⇒ (d) by 16.13 and 17.1. (d) =⇒ (c) and (c) =⇒ (b) are obvious. Proposition 17.20 Every AN R band (X, c) is simple homotopy equivalent to one such that c : X−→R is proper homotopic to a proper approximate ﬁbration. The proof of 17.20 will be based on the following two lemmas. Lemma 17.21 Suppose M is a ﬁnitely dominated manifold such that ∂M is also ﬁnitely dominated. Then there exist a compact subset C ⊆ M and a homotopy h : idM h1 : M × I−→M such that : (i) h1 (M ) ⊆ C , (ii) if x ∈ ∂M (resp. int(M )) then h(x × I) ⊆ ∂M (resp. int(M )) . Proof This is a standard construction using a collar of ∂M in M . Lemma 17.22 Let (N, ∂N ) be a compact n-dimensional manifold with boundary such that π1 (∂N )−→π1 (N ) is a split injection. If (b, ∂b) : (N, ∂N ) −→S 1 is a map such that (N, b) is a band then the boundary (∂N, ∂b) is also a band. Proof Let N be the universal cover of N , and let ∂N be the corresponding cover of ∂N . We need to show that the inﬁnite cyclic cover ∂N = (∂b)∗ R of ∂N is ﬁnitely dominated, which by 6.9 (i) is equivalent to the Z[π1 (N )]ﬁnite domination of the cellular Z[π1 (N )]-module chain complex C(∂N ). The inﬁnite cyclic cover N = b∗ R of N is ﬁnitely dominated, so that C(N ) is Z[π1 (N )]-ﬁnitely dominated, and so is the n-dual Z[π1 (N )]-module chain complex C(N )n−∗ . By the exactness of 0 −→ C(∂N ) −→ C(N ) −→ C(N , ∂N ) −→ 0 and the Poincar´–Lefschetz Z[π1 (N )]-module chain equivalence e C(N , ∂N ) C(∂N ) C(N )n−∗ there is deﬁned a Z[π1 (N )]-module chain equivalence
C(C(N )−→C(N )n−∗ )∗+1 ,

Proof of 17.20 By West’s result on the homotopy ﬁniteness of compact AN R’s there is no loss of generality in assuming that (X, c) is a CW band. Let N be a regular neighbourhood of X in some Euclidean space of suﬃciently high dimension that the inclusion ∂N −→N induces an isomorphism π1 (∂N ) ∼ π1 (N ) and dim(N ) > 5. = Thus, X is simple homotopy equivalent to N and there is a map b : N X −→ S 1
c

inducing a ﬁnitely dominated inﬁnite cyclic cover N . It follows from Proposition 17.14 that the induced map b : N −→R is a proper bounded ﬁbration. Lemma 17.22 implies that ∂N is also ﬁnitely dominated, so ∂b : ∂N −→R is also a proper bounded ﬁbration. The rest of the proof consists of applying a stratiﬁed sucking principle from Hughes [74] to show that b is boundedly homotopic to an approximate ﬁbration. (Note that 16.10 cannot be used because N has a boundary.) The idea is that N is a stratiﬁed space with strata ∂N and int(N ). The proof of Proposition 17.14 actually shows that (b, ∂b) : (N , ∂N )−→R is a proper stratiﬁed bounded ﬁbration. This is because Lemma 17.22 shows that (N , ∂N ) is ﬁnitely dominated in a stratiﬁed sense. Now use the stratiﬁed sucking theorem [74]. Remark 17.23 (i) The wrapping up (X, d) of a manifold approximate ﬁbration (X, d) can also be constructed by the end obstruction theory of Siebenmann [140] (quoted in 10.2) and the projective surgery theory of Pedersen and Ranicki [109], as follows. The total projective surgery obstruction groups Sp (K) of [109] are deﬁned ∗ for any space K to ﬁt into the algebraic surgery exact sequence . . . −→ Hm (K; L. ) −→ Lp (Z[π1 (K)]) −→ Sp (K) m m −→ Hm−1 (K; L. ) −→ . . . , with L. the 1-connective simply-connected surgery spectrum such that π∗ (L. ) = L∗ (Z), and Ap the assembly map in projective L-theory. The total projective surgery obstruction sp (K) ∈ Sp (K) of a ﬁnitely m dominated m-dimensional Poincar´ space K is such that sp (K) = 0 if e (and for m ≥ 5 only if) K × S 1 is homotopy equivalent to a compact (m + 1)-dimensional manifold L. (See Ranicki [125] for a detailed exposition of the total surgery obstruction.) If sp (K) = 0 then the composite c : L K × S 1 −→S 1 deﬁnes an (m + 1)-dimensional manifold band (L, c), such that K is homotopy equivalent to the inﬁnite cyclic cover L = c∗ R of L. It was shown in [109] that L can be chosen such that L×S 1 is homeomorphic to L × R.
Ap

with i = inclusion : M −→W . We now show that the torsion of (W ; M, M ) is well-deﬁned, by which we mean that it is independent of the choices made in constructing M . So suppose Gs : M −→M is another isotopy such that cGs is (a1 + 2a2 )-close to gs c. By a one-parameter and relative version of Approximate Isotopy Covering 17.4 (see Hughes, Taylor and Williams [77]) there is a two-parameter isotopy Γs,t : M −→M such that Γ0,t = idM , cΓs,t is (a1 + 2a2 )-close to gs c, Γs,0 = Gs and Γs,1 = Gs . The homeomorphism β = (ζ × idI )Γ1,− : M × 1 × I −→ M × 1 × I deﬁnes a properly discontinuous action of Z on M × I (Proposition 17.5). We claim that the natural projection (M × I)/β−→I is a locally trivial bundle projection. First note that since M × I−→(M × I)/β is a covering and the composition M × I−→(M × I)/β−→I is locally trivial, it follows easily that (M × I)/β−→I is a Serre ﬁbration. Because (M × I)/β and I are ﬁnite dimensional AN R’s, (M × I)/β−→I is also a Hurewicz ﬁbration (Ungar [162]). The ﬁbres are manifolds of dimension greater than 4, so that it is locally trivial (Chapman and Ferry [29]). Since β0 = ζG1 , there is a trivializing homeomorphism α : M × I−→(M × I)/β such that α0 = idM . Let H : (M × I)/β × R −→ M × I × R be the homeomorphism given by the composite (M × I)/β × R ∼ T (β) ∼ T (ζ × idI ) ∼ M × I × R. = = = Since H0 is isotopic to H1 ◦ (α1 × idR ), it follows that the h-cobordisms determined by H0 and H1 ◦ (α1 × idR ) are homeomorphic (using the Isotopy Extension Theorem of Edwards and Kirby [41]), and have the same torsion. Hence, the h-cobordisms determined by H0 and H1 have the same torsion. These are the h-cobordisms given by the two sets of data so we have established well-deﬁnedness. This also shows that the torsion depends only on the homotopy class of c, for if c c1 , then both c and c1 induce data for constructing the relaxation of (M, c) yielding the same torsion.

218

Ends of complexes

(a) =⇒ (b) We now assume that c is homotopic to a manifold approximate ﬁbration and show that the h-cobordism is trivial. By the preceding paragraph we may assume that c itself is a manifold approximate ﬁbration. Therefore, in the construction of M we may take d = c, a3 > 17 and a1 , a2 as small as we like. Then we shall have cGs a1 -close to gs c. Proposition 17.5 can be used to show that ζGs acts properly discontinuously on M for each s. Then the homeomorphism γ = (ζ × idI )G : M × I −→ M × I induces a properly discontinuous action on M × I with (M × I)/γ ∼ M × I = by the argument above. Since γ0 = ζ and γ1 = ζG1 , it follows as above that γ may be used to show that the h-cobordism between M = M /ζ and M = M /ζG1 is trivial. (b) =⇒ (c) If h : M × I−→W is a homeomorphism with h0 = idM then M = M × {1}−→M ⊆ W is a homeomorphism homotopic to f . (c) =⇒ (a) Since c c f , if f is homotopic to a homeomorphism h : M −→M then c is homotopic to the manifold approximate ﬁbration c f . Remark 18.4 (i) For any (n + 1)-dimensional h-cobordism (W ; M, M ) the torsion of the homotopy equivalence f : M −→W −→M is τ (f ) = τ (M −→W ) − τ (M −→W ) = τ (M −→W ) + (−)n τ (M −→W )∗ ∈ W h(π1 (M )) . It will be shown in 26.13 that for the relaxation h-cobordism (W ; M, M ) of 18.3 (v) τ (M −→W ) and τ (M −→W ) are in complementary direct summands of the Whitehead group of π1 (M ) = π1 (M ) ×ζ∗ Z, namely the two copies of the reduced nilpotent class group Nil0 , and that there is a Poincar´ e n−1 τ (M −→W )∗ . Thus the conditions (a), (b), duality τ (M −→W ) = (−) (c) in 18.3 are also equivalent to : (d) τ (f ) = 0 . In particular, f is simple if and only if f is homotopic to a homeomorphism. (ii) It will follow from 26.10 (ii) that an n-dimensional manifold band (M, c) with n ≥ 5 is relaxed if and only if the homotopy equivalence f : M −→M in 18.3 (v) is simple. Combining this with (i) and 18.3 (iii) gives : a manifold band (M, c) with dim M ≥ 5 is relaxed if and only if the map c : M −→S 1 is homotopic to a manifold approximate ﬁbration.
h|

−1 The map is well-deﬁned for if x ∈ U then ζG1 (x) = f+ ζf− (x). To show that the map is onto it suﬃces to show that every x ∈ U− is ∼-related to a point in V . This is clear for d(x) > −1, so we consider the case d(x) ≤ −1. −1 −1 For such an x we have x ∼ f+ ζf− (x) = f+ ζ(x), so it suﬃces to show that −1 f+ ζ(x) ∈ V . For this we must make sure that a1 is chosen small enough that a3 − a1 > 2a2 , in which case

The construction of (X(h), d(h)) from (X, d), h is a homotopy theoretic version of Siebenmann twist glueing (15.17). If h : (X, d)−→(X, d) is a proper homotopy equivalence which is either an end-preserving covering translation or the identity we reﬁne the construction of (X(h), d(h)), F (h) to obtain a relaxed CW π1 -band (X[h], d[h]) in the homotopy type of (X(h), d(h)) with an inﬁnite simple homotopy equivalence F [h] : (X, d) −→ (X[h], d[h]) . In Chapter 20 we shall show that the wrapping up (X, d) of a manifold ribbon (X, d) constructed in Chapter 17 has the simple homotopy type of the 1-twist glueing (X[1], d[1]), and that the relaxation (M , c ) of a manifold band (M, c) constructed in Chapter 18 has the simple homotopy type of the ζ-twist glueing (X[ζ], d[ζ]) of the manifold ribbon (X, d) = (M , c). 222

Similarly for A[z −1 ], which is isomorphic to A[z]. (ii) The Laurent polynomial extension of A is the ring A[z, z −1 ] = {
∞ j=−∞

aj z j | aj ∈ A, {j ∈ Z | aj = 0} ﬁnite}

obtained from A[z] by inverting z. Remark 21.2 (i) Given a ring A and an automorphism α : A−→A let z be an indeterminate over A such that az = zα(a) (a ∈ A) . Given an A-module P let α! P be the A-module with elements α! x (x ∈ P ) 255

can be cut open along either C or D, so that both W(f (i, j)+ , f (i, j)− ) and W(g(i, j)+ , g(i, j)− ) are chain equivalent to B.

22
Algebraic bands

An ‘algebraic band’ is the chain complex analogue of a CW band. We shall now recall from Ranicki [124, Chapter 20] the algebraic band version of the Whitehead torsion obstruction of Farrell [47] and Siebenmann [145] for ﬁbring a manifold band over S 1 . In Chapters 23–25 we shall develop the chain complex analogues of forward and reverse tameness, relaxation, and ribbons, which will then be applied in Chapter 26 to obtain an algebraic version of the homotopy theoretic twist glueing of Chapter 19. Deﬁnition 22.1 A chain complex band is a ﬁnite based f.g. free A[z, z −1 ]module chain complex C which is A-ﬁnitely dominated, so that the projective class [C] ∈ K0 (A) is deﬁned. Proposition 22.2 (i) If C is a ﬁnite based f.g. free A-module chain complex and h : C−→C is a chain equivalence the algebraic mapping torus T (h) is an A[z, z −1 ]-module chain complex band. (ii) If C is a ﬁnitely dominated A-module chain complex and h : C−→C is a chain equivalence then any ﬁnite based f.g. free A-module chain complex E chain equivalent to T (h) is an A[z, z −1 ]-module chain complex band. If (D, f : C−→D, g : D−→C, gf 1 : C−→C) is a ﬁnite domination of C then E = T (f hg : D−→D) is such a chain complex band in the canonical simple chain homotopy type of T (h). For any f.g. free A[z, z −1 ]-module chain complex C there are deﬁned exact sequences of A[z, z −1 ]-module chain complexes 0 −→ C[z, z
−1

We shall now develop algebraic analogues of tameness for A[z]- and A[z, z −1 ]module chain complexes for any ring A, corresponding to the geometric tameness properties of the ends of inﬁnite cyclic covers of ﬁnite CW complexes. The algebraic theory of tameness will be applied in 23.22 to prove + that an end W of an inﬁnite cyclic cover W of a ﬁnite CW complex W with π1 (W ) = π1 (W ) × Z is forward (resp. reverse) tame if and only if the cellular Z[π1 (W )]-module chain complex C(W + ) is forward (resp. reverse) tame. Deﬁnition 23.1 (i) The formal power series extension of A is the ring
∞

A[[z]] = {
j=0

aj z j | aj ∈ A} ,

without any ﬁniteness conditions on the coeﬃcients aj . Similarly for A[[z −1 ]], which is isomorphic to A[[z]]. (ii) The Novikov polynomial extension of A is the ring A((z)) = A[[z]][z −1 ] = {
∞ j=−∞

be a polynomial over A, with am , an = 0 ∈ A. (i) The polynomial p(z) is a unit in A((z)) (resp. A((z −1 ))) if and only if am (resp. an ) is a unit in A. (ii) The 1-dimensional f.g. free A[z]-module chain complex C + = C(p(z) : A[z]−→A[z]) is forward (resp. reverse) tame if and only if am (resp. an ) is a unit in A. (iii) The 1-dimensional based f.g. free A[z, z −1 ]-module chain complex C = C(p(z) : A[z, z −1 ]−→A[z, z −1 ]) is a band if and only if am , an are both units in A. Example 23.20 For any ring A and central non-zero divisor s ∈ A the localization of A inverting s and the s-adic completion of A (2.28) are such that A[1/s] = A[z]/(1 − zs) , As = lim (A/sk A) = A[[z]]/(z − s) . ← −
k

of [28]) and Φ+ (W, c) = 0 (the second obstruction of [28]). In both [27] and [28] it was assumed that W is homotopy ﬁnite. If (K, φ : W K) is a ﬁnite structure on W then K is a ﬁnite CW complex with a self homotopy equivalence h = φζφ−1 : K−→K, and there is deﬁned a homotopy equivalence T (h)−→W , such that τ (r) = i∗ τ (h) , Φ+ (W, c) = τ (T (h)−→W ) ∈ W h(π × Z) with i∗ : W h(π)−→W h(π × Z) the inclusion. However, it is not necessary to assume that W is homotopy ﬁnite in the reformulation.

25
Algebraic ribbons

‘Algebraic ribbons’ are the chain complex analogues of the geometric ribbons of Chapter 15. We shall now develop the algebraic theory of ribbons, in the context of the bounded algebra of Pedersen and Weibel [110] and Ranicki [124]. A chain complex ribbon is a ﬁnite chain complex C in the category CR (A) of R-bounded A-modules (for some ring A) with the end properties of a chain complex band. In Chapter 26 we shall use algebraic ribbons to develop the algebraic theory of twist glueing. An A[z, z −1 ]-module chain complex band is an example of a chain complex ribbon; in 26.6 it will be shown that every chain complex ribbon C is simple chain equivalent to a chain complex band C, the ‘wrapping up’ of C. In Chapter 27 we shall describe the eﬀects of wrapping up in algebraic K- and L-theory. We refer to Ferry and Pedersen [58] and Ranicki [124] for accounts of bounded topology and algebra, only repeating the most essential deﬁnitions here. Deﬁnition 25.1 Let A be a ring, and let B be a metric space. The Bbounded A-module category CB (A) is the additive category with objects B-graded A-modules M =
x∈B

A B-bounded CW complex (X, d) is a CW complex X with a proper cellular map d : X−→B such that the diameters of the images in B of the cells e ⊂ X are uniformly bounded, that is there exists a bound b ≥ 0 with diameter(d(e)) < b for all e ⊂ X. If X is a regular cover of X with group of covering translations π the cellular Z[π]-module chain complex C(X) is deﬁned in CB (Z[π]). A B-bounded map f : (X, d)−→(Y, e) of B-bounded CW complexes is a proper cellular map f : X−→Y such that there exists a bound b ≥ 0 with dB (e(f (x)), d(x)) ≤ b (x ∈ X) . A B-bounded map f induces a chain map f : C(X)−→C(Y ) in CB (Z[π]) for any regular cover Y of Y with group of covering translations π, with X = f ∗ Y the pullback cover of X. A B-bounded homotopy equivalence f : (X, d)−→(Y, e) of B-bounded CW complexes has a B-bounded torsion τB (f ) = τB (f : C(X)−→C(Y )) ∈ W hB (Z[π1 (X)]) . A B-bounded h-cobordism has B-bounded torsion in W hB (Z[π1 ]), and there are bounded versions of the h- and s-cobordism theorems; in the bounded version of the Wall surgery theory a B-bounded normal map has a surgery obstruction in L∗ (CB (Z[π1 ])) (Ferry and Pedersen [58]). For bounded surgery theory A is a ring with involution (e.g. a group ring), and CB (A) is an additive category with involution, so that the L-groups L∗ (CB (A)) are deﬁned as in Ranicki [124]. Deﬁnition 25.3 Let A be a ring. (i) Let M(A) be the additive category of A-modules. Deﬁne the sum and product functors : CB (A) −→ M(A) ; M −→ M =
x∈B

M (x) , M (x) .
x∈B

: CB (A) −→ M(A) ; M −→ M

lf

=

The inclusion i : M −→M lf deﬁnes a natural transformation from to . (ii) The end complex of a chain complex C in CB (A) is the A-module chain complex e(C) = C(i : C−→C lf )∗+1 .

302

Ends of complexes

Example 25.4 Let (X, d) be a connected B-bounded CW complex, so that the cellular chain complex C(X) of the universal cover X of X is a ﬁnite chain complex in CB (Z[π]) with π = π1 (X). The locally π-ﬁnite cellular chain complex of X (5.5 (i)) is C lf,π (X) = C(X)lf . Let e(X) = p∗ X be the cover of the end complex e(X) obtained from X by pullback along the projection p : e(X)−→X; ω−→ω(0). If X is forward tame the Z[π]-module chain complex at ∞ C ∞,π (X) is homology equivalent to the end complex e(C(X)) of C(X). Deﬁnition 25.5 A subobject M ⊆ M of an object M in CB (A) is the object M = M (B ) =
y∈B

A.1. Regular covers and singular homology and locally ﬁnite singular homology with local coeﬃcients In this section W denotes a path-connected space. Let p : W −→W be a regular cover with group of translations π. It is well-known that the ordinary singular chain complex S(W ) is a chain complex of Z[π]-modules and has an interpretation in terms of local coeﬃcients. Namely, S(W ) is isomorphic to S∗ (W ; Γ) where Γ is a local system (described below) with Γx ∼ Z[π] for each x ∈ W and S∗ (W ; Γ) is the singular = chain complex of W with coeﬃcients in the local system Γ (see Whitehead [169, p. 278] and Spanier [150, p. 179]). The goal here is to generalize this to the locally ﬁnite case. However, when lf S∗ (W ; Γ) is replaced by the locally ﬁnite S∗ (W ; Γ) we get a chain complex isomorphic to S lf,π (W ), the locally π-ﬁnite singular chain complex, rather than S lf (W ). This should help convince the reader that S lf,π (W ) is a more natural object than S lf (W ). We begin by setting up the notation needed to describe the local Z[π] coeﬃcient system on W . Assume that W and W are path-connected. For each x ∈ W ﬁx x ∈ ˜ p−1 (x) and let πx = π1 (W, x)/p∗ π1 (W , x) . ˜ Of course, πx ∼ π for each x ∈ W , but we need to have an explicit isomor= phism γx on hand. To this end, for each x ∈ W deﬁne a bijection αx : π −→ p−1 (x) ; g −→ g(˜) x 325

326 and also a bijection

Ends of complexes

βx : p−1 (x) −→ πx ; y −→ [pω] where ω is a path in W from x to y. Let ˜ γx = βx ◦ αx : π −→ πx . Then γx is a group isomorphism. Given a path ω : I−→W deﬁne an isomorphism πω : πω(0) −→ πω(1) ; [λ] −→ [ω −1 ∗ λ ∗ ω] . Let Γx = Z[πx ] for x ∈ W . The isomorphism γx : π −→ πx induces a ring isomorphism, also denoted by γx , γx : Z[π] −→ Γx , which thereby gives Γx the structure of a Z[π]-module. For each path ω : I−→W πω extends to an isomorphism Γω : Γω(0) −→ Γω(1) , so that there is deﬁned a local system Γ of Z[π]-modules on W [150, p. 179]. Using the local system Γ we can construct the usual singular chain complex of W with coeﬃcients in Γ as well as the locally ﬁnite singular chain complex of W with coeﬃcients in Γ. Let S∗ (W ; Γ) = {Sq (W ; Γ), ∂} be the singular chain complex of W with q local Z[π]-coeﬃcients as deﬁned in [150, p. 179]. For σ ∈ W ∆ , Γ(σ) is the Z[π]-module of Γ sections of σ. An element of Γ(σ) is a function s : ∆q −→
y∈∆q

where the direct sum is over the set of singular q-simplexes in W . On the lf,π other hand, Sq (W ) is not a free Z[π]-module, and it is not even clear when it is a non-trivial direct product. Proposition A.1 There are isomorphisms of Z[π]-module chain complexes
lf,π lf A : S∗ (W ) −→ S∗ (W ; Γ) , Alf,π : S∗ (W ) −→ S∗ (W ; Γ) .

where as above τ is the unique lift of τ with τ (0) = τ (0). ˜ ˜ These formulas generalize easily to the locally π-ﬁnite case as follows. An lf,π element of Sq (W ) can be uniquely written as rτ τ ˜
q τ ∈W ∆

Clearly, the function σ−→δσ (rσ ) has the correct local ﬁniteness property and an inverse B lf,π for Alf,π can be deﬁned similarly to B.

Appendix A. Locally ﬁnite homology with local coeﬃcients

329

Remarks A.2 (i) It might be worthwhile to note how the local system Γx depends on the choice of the basepoints x ∈ p−1 (x). First note that the ˜ group πx is independent of x, but γx is not. For if x ∈ p−1 (x) is another ˜ choice inducing the isomorphism γx : π−→πx , let λx be a path in W from ˆ x to x . If λx : πx −→πx is the inner automorphism given by ˜ ˆ λx (a) = [pλx ] ∗ a ∗ [pλx ]−1 for each a ∈ πx , ˆ then γx = λx ◦ γx . Since the isomorphism γx : Z[π]−→Γx depends on the choice of x, so does the Z[π]-module structure on Γx . In fact a diﬀerent ˜ choice of basepoints, say x ∈ p−1 (x), induces a diﬀerent local system Γ which need not be isomorphic to Γ (in the sense of Steenrod [157]). However, ˆ ˆ the inner automorphisms λx : πx −→πx induce isomorphisms λx : Γx −→Γx ˆ which in turn induce isomorphisms of chain complexes λx : S∗ (W ; Γ) ∼ = S∗ (W ; Γ ). (ii) In order to introduce another local system Λ on W consider the Z[π]module Z[[π]]. It has elements written as formal products g∈π ng g with ng ∈ Z. Addition is deﬁned termwise and Z[π] acts on Z[[π]] via mh h ·
h∈π g∈π

A.3. Locally ﬁnite cellular homology with local coeﬃcients Locally ﬁnite cellular homology with global coeﬃcients is deﬁned by Geoghegan [63] where it is called inﬁnite cellular homology. We need to generalize [63] to the case of local coeﬃcients. Let W and Γ be as in section A.2 above, but now assume that W is locally ﬁnite. Deﬁne the nth locally ﬁnite cellular chain module of W with local coeﬃcients Γ by
lf Cn (W ; Γ) = α∈In

when W is a strongly locally ﬁnite CW complex (the module on the right is the locally ﬁnite singular homology module of W with local coeﬃcients lf Γ, namely Hn (S∗ (W ; Γ))). A.4. Local Z[π]-coeﬃcients and cellular homology Let Γ be the local system of Z[π] coeﬃcients on the connected locally ﬁnite CW complex W as described in section A.1 above. Note that there is a commutative diagram
S∗ (W n−1 ) A
∗

Proof (i) is just the local coeﬃcient version of 3.16. It is also derived by Spanier [150, Theorem 9.12]. (ii) is clear from the direct sum and direct product descriptions above. In the constant coeﬃcient case it was also observed by Geoghegan [63].

Path spaces are widely used in topology, notably in homotopy theory and Morse theory. They have also been long used to describe the local topological behaviour of spaces. In 1955 John Nash [102] used path spaces to give a homotopy theoretic model of the tangent space of a smooth manifold. Given a smooth manifold M and x ∈ M , Nash considered the path spaces T (M, x) = {ω : I−→M | ω(0) = x , ω(t) = x for all t > 0} , T (M ) =
x∈M

T (M, x) ⊆ M I .

If dim(M ) = m then each T (M, x) is homotopy equivalent to S m−1 , and the map T (M ) −→ M ; ω −→ ω(0) is a ﬁbration with ﬁbre S m−1 . Nash observed that this ﬁbration is ﬁbre homotopy equivalent to the tangent sphere bundle of M . Consequently, the ﬁbre homotopy type of the tangent sphere bundle of a smooth manifold depends only on the underlying topological type of the manifold. This observation constituted Nash’s proof of Thom’s theorem on the topological invariance of the Stiefel–Whitney classes. Hu [69] studied the path space T (X, x) where X is an arbitrary space. He used the algebraic topology of T (X, x) to deﬁne the local algebraic topology of X at x. Later, Hu [70, 71] considered a related construction to model the normal, as opposed to the tangential, direction. Given a subspace Y ⊆ X and a point y ∈ Y , let N (X, Y, y) = {ω : I−→X | ω(0) = y, ω(t) ∈ Y for all t > 0} , / N (X, Y ) =
y∈Y

N (X, Y, y) ⊆ X I .

In certain very special cases Hu showed that evaluation at 0 deﬁnes a ﬁbration N (X, Y )−→Y . 335

336

Ends of complexes

Fadell [43] used Hu’s generalization to give a homotopy theoretic model of the normal bundle of a submanifold P ⊂ M without assuming a smooth structure on M . In [43] it is proved that for a p-dimensional locally ﬂat topological submanifold P of an m-dimensional topological manifold M , the evaluation at 0 deﬁnes a ﬁbration N (M, P )−→P with ﬁbre S m−p−1 . If M is a smooth manifold and P is a smooth submanifold, then N (M, P )−→P is ﬁbre homotopy equivalent to the normal sphere bundle of P in M . Consequently, the ﬁbre homotopy type of the normal sphere bundle depends only on the underlying topological type. In fact, every ﬁnite CW complex P can be embedded in M = S m (m large) with a regular neighbourhood (W, ∂W ). If m − p ≥ 3 then P is a p-dimensional Poincar´ complex if and e only if the homotopy ﬁbre of the inclusion ∂W −→W is S m−p−1 , in which case S m−p−1 −→ N (S m , P ) ∂W −→ P W

is the normal ﬁbration of Spivak [152], which depends only on the homotopy type of P . The path space constructions of Nash, Hu, and Fadell gave homotopy theoretic models for tangent and normal sphere bundles for topological manifolds, even when those manifolds have no smooth structure. Later, Milnor [98] invented microbundles as a better substitute for the tangent bundle of a topological manifold. Then Kister [85] and Mazur used that theory to construct a well-deﬁned tangent bundle (rather than just a microbundle) of a topological manifold. On the other hand, for a locally ﬂat topological submanifold P of a topological manifold M , Rourke and Sanderson [138] showed that P need not have a normal bundle neighbourhood in M . Thus, Fadell’s construction has stood as the best model for a normal bundle in the general topological setting. For more recent results in this direction, see Hughes, Taylor and Williams [78]. Quinn [116] used path space models in his work on stratiﬁed spaces. In order for a pair (X, Y ) to be a homotopically stratiﬁed set in the sense of [116], it is necessary that the evaluation map N (X, Y )−→Y be a ﬁbration, although in this generality the ﬁbre need not be a sphere. N (X, Y ) is called the homotopy link of Y in X in [116] (see 12.11 for the deﬁnition). The homotopy model for the behaviour at inﬁnity of a space W is the Nash (spherical) tangent space T (W ∞ , ∞) of the one-point compactiﬁcation W ∞ at ∞. Of course, T (W ∞ , ∞) is the same space as the Hu–Fadell normal space N (W ∞ , {∞}). Quinn would consider W ∞ to be a stratiﬁed set with two strata, W and {∞}. Then T (W ∞ , ∞) is the homotopy link of {∞} in W ∞ . All of these points of view yield the same object, namely the end

Appendix B. A brief history of end spaces

337

space e(W ). Freedman and Quinn [60, p. 214] call e(W ) a homotopy collar of W . A diﬀerent point of view which often arises in studying the end theory of W is to focus attention on the inverse system of complements of compact subsets of W . When W can be written as an ascending union
∞

K1 ⊆ K2 ⊆ K3 ⊆ . . . ⊆
i=1

Ki = W

of compact subspaces, one considers the inverse sequence of inclusions W \K1 ← − W \K2 ← − W \K3 ← − . . . . − − − (This is the point of view of Porter [111], for example.) The homotopy inverse limit of this sequence is homotopy equivalent to the end space e(W ). This model for e(W ) has been exploited by Edwards and Geoghegan [39], and by Edwards and Hastings [40] for problems related to proper homotopy theory, shape theory, and end theory.

Appendix C. A brief history of wrapping up

In this appendix we use wrapping up to denote the geometric compactiﬁcation procedure which passes from a non-compact space X with a proper map X−→Rn to a compact space X with a map X−→T n . The wrapping up of Chapter 17 is the special case n = 1. Wrapping up sometimes goes under the name ‘belt buckle trick’, ‘furling’ and it is a special case of the ‘torus trick’. In this appendix we shall give a brief history of the development and applications of wrapping up. In the applications, it is also frequently useful to consider the passage in the reverse direction. In 1964 M. Brown (unpublished) proved that if X, Y are compact Hausdorﬀ spaces which are related by a homeomorphism f : X × R−→Y × R then X ×S 1 , Y ×S 1 are related by a homeomorphism f : X ×S 1 −→Y ×S 1 . Three proofs are available in the literature : Siebenmann [142, 145] and Edwards and Kirby [41]. For manifold X, Y the result was obtained by h-cobordism theory (17.3). ‘Novikov ﬁrst exploited a torus furling idea in 1965 to prove the topological invariance of rational Pontrjagin classes. And this led to Sullivan’s partial proof of the Hauptvermutung. Kirby’s unfurling of the torus was a fresh idea that proved revolutionary.’ (Siebenmann [146, footnote p. 135]). Edwards and Kirby [41] used wrapping up to prove the local contractibility of the homeomorphism group of a compact manifold. Siebenmann [145] developed a general twist glueing construction, with wrapping up as a special case, and used it to analyse the the obstruction to ﬁbring manifolds over S 1 (cf. Chapter 17). Chapman [25, 26] used this form of wrapping up to obtain approximation results for manifolds, such as the sucking principle (16.13). Hughes [72] developed a parametrized wrapping up, which led to the classiﬁcation of manifold approximate ﬁbrations by Hughes, Taylor and Williams [77]. 338

Appendix C. A brief history of wrapping up

339

Various forms and applications of geometric wrapping up have also appeared in the works of Anderson and Hsiang [2], Bryant and Pacheco [17], Burghelea, Lashof and Rothenberg [18], Ferry [53], Freedman and Quinn [60], Madsen and Rothenberg [88], Prassidis [112], Rosenberg and Weinberger [137], Steinberger and West [158], Weinberger [166], Weiss and Williams [167], . . . .